Computer Modeling of Matter

Computer Modeling of Matter

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Computer Modeling of Matter Peter Lykos, EDITOR Illinois Institute of Technology

Based on a symposium sponsored by the ACS Division of Computers in Chemistry at the 175th Meeting of the American Chemical Society, Anaheim, California, March 14,

1978.

ACS SYMPOSIUM SERIES 86

AMERICAN CHEMICAL SOCIETY WASHINGTON, D. C.

1978

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Library of Congress CIP Data Computer modeling of matter. (ACS symposium series: 86 ISSN 0087-6156) Includes bibliographies and index. 1. Molecular theory—Data processing—Congresses. 2. Matter—Properties—Data processing—Congresses. I. Lykos, Peter George. II. American Chemical Society. Division of Computers in Chemistry. III. Series: American Chemical Society. ACS symposium series; 86. QD461.C632 542'.8 78-25828 ISBN 0-8412-0463-2 ASCMC 8 86 1-272 1978

Copyright © 1978 American Chemical Society All Rights Reserved. The appearance of the code at the bottom of thefirstpage of each article in this volume indicates the copyright owner's consent that reprographic copies of the article may be made for personal or internal use or for the personal or internal use of specific clients. This consent is given on the condition, however, that the copier pay the stated per copy fee through the Copyright Clearance Center, Inc. for copying beyond that permitted by Sections 107 or 108 of the U.S. Copyright Law. This consent does not extend to copying or transmission by any means—graphic or electronic—for any other purpose, such as for general distribution, for advertising or promotional purposes, for creating new collective works, for resale, or for information storage and retrieval systems. The citation of trade names and/or names of manufacturers in this publication is not to be construed as an endorsement or as approval by ACS of the commercial products or services referenced herein; nor should the mere reference herein to any drawing, specification, chemical process, or other data be regarded as a license or as a conveyance of any right or permission, to the holder, reader, or any other person or corporation, to manufacture, reproduce, use, or sell any patented invention or copyrighted work that may in any way be related thereto. PRINTED IN ΤΗE UNITED STATES OF AMERICA

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

ACS Symposium Series Robert F. Gould,

Editor

Advisory Board Kenneth B. Bischoff

Nina I. McClelland

Donald G . Crosby

John B. Pfeiffer

Jeremiah P. Freeman

Joseph V . Rodricks

E. Desmond Goddard

F. Sherwood Rowland

Jack Halpern

Alan C. Sartorelli

Robert A . Hofstader

Raymond B. Seymour

James P. Lodge

Roy L. Whistler

John L. Margrave

Aaron Wold

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

FOREWORD The ACS SYMPOSIUM SERIES was founded in 1974 to provide

a medium for publishin format of the Series parallels that of the continuing ADVANCES IN CHEMISTRY SERIES except that in order to save time the papers are not typeset but are reproduced as they are submitted by the authors in camera-ready form. Papers are reviewed under the supervision of the Editors with the assistance of the Series Advisory Board and are selected to maintain the integrity of the symposia; however, verbatim reproductions of previously published papers are not accepted. Both reviews and reports of research are acceptable since symposia may embrace both types of presentation.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

PREFACE he ever increasing power of computers and the continuing decrease in their cost enable the chemist to construct increasingly sophisticated mathematical models of bulk matter—both at equilibrium and changing in time—from a molecular perspective. Thanks to the pioneering work of Bernie Alder and others who developed the method of molecular dynamics and to the Monte Carlo method of Metropolis used for equilibrium data, as the size and speed of computers increased i Frank Stillinger (/. Chem. of molecular dynamics to the most important and complex bulk matter of all—liquid water. That seminal paper sparked a great interest in modeling on the part of chemists. The important discovery that a mathematical model whereby one averages the individual properties of a few hundred interacting molecules suffices to assess the bulk properties of many important systems suggests that chemists now can build more effective bridges between atomic and molecular physics on the one hand and surface chemistry and chemical thermodynamics and kinetics on the other. In the recent "Science Update: Physical Chemistry" in Chemical and Engineering News ((1978) June 5, p 20-21), the impact of the "computer primarily as an aid to modeling was highlighted as the most pervasive and important factor influencing physical chemistry today. In that article Mitch Waldrop wrote, "So important have the big computers become that number crunching and theoretical chemistry often seem synonymous. The applications can be divided into three broad areas: quantum chemistry, chemical dynamics, and statistical mechanics. If computers were big enough, those three would form a logical sequence for the complete ab initio calculation of the properties of bulk matter." The number and range of computer-based models of bulk matter has increased rapidly. An important international conference (with proceedings), "Computational Physics of Liquids and Solids" held April, 1975, at Queen's College in Oxford, involved 34 papers that displayed a broad range of chemical phenomena being studied in this manner. Just a year and a half later the Faraday Division of the Chemical Society, London, held a two-day symposium on "Newer Aspects of Molecular Relaxation Processes" at the Royal Institution, London. The so-called 'experimental technique of examining motions in a computer-generated ix In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

liquid* was considered together with experimental methods, with theoretical models of relaxation processes, and with the hydrodynamics of rotation in fluids. Two books have appeared which, while not presenting a unified view of the field nor a critical assessment of the literature, do provide the interested scientist with an entree to the use of computer simulation of liquids ("Theory of Simple Liquids," J. P. Hansen, I. R. McDonald, Academic, 1976; "Atomic Dynamics in Liquids," N . H . March, M . P. Tosi, Halsted (Wiley), 1977). The impact on statistical mechanics as a discipline is reflected in the two-volume work, "Statistical Mechanics," B. J. Berne, Ed., Part A, Equilibrium Techniques, and Part B, TimeDependent Processes, Plenum, 1977. Indeed, in his Nobel prize address, I. Prigogine made several references to the use of computer simulations as an aid to developmen macroscopic and microscopic aspects of the second law of thermodynamics (Science (1978) 201,777-785). Enough experience has been gained through the design and testing of such models that chemists interested in gaining insight into particular chemical systems are beginning to apply the models in a rather sophisticated manner. For example, William Jorgensen, a theoretical organic chemist interested in solvation effects in organic chemistry, has begun by looking at liquid hydrogen fluoride in a paper to be published in December, 1978, /. Am. Chem. Soc. Indeed theorists are organizing and presenting their theories with careful attention to how they might be applied using a computer (for example, see "Simulation of polymer dynamics 1. General theory and 2. Relaxation rates and dynamic viscosity," M. Fixman, /. Chem. Phys. (1978) 69, p 1527, 1538). The call for contributions to this symposium has resulted in an interesting collection of eighteen chapters which provide the reader with a variety of touchstones ranging from the first chapter, a straight-forward application by K. Heinzinger of the Rahman-Stillinger method to an aqueous solution of sodium chloride, to the last chapter, an overview from microphysics to macrochemistry via discrete simulations including chemical kinetics. The other chapters include examination of an algorithm assessing the importance of three-body interactions and of algorithms reducing machine requirements with regard to size of store and speed. Most of the work done to date has been based on classical mechanics. The chapter by M . H . Kalos brings home the fact that one needs to examine the validity of such models from within the framework of quantum mechanics. The interesting phenomenon of collective modes of motion is exemplified by M . L . Klein's chapter on collective modes in x In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

solids. The liquid-vapor interface is addressed in two chapters, one by Rao and Berne and the other by Thompson and Gubbins. Sundheim's chapter on high field conductivity emphasizes that the modeler is free to impose whatever simulated physical conditions he pleases and, in fact, can achieve in his computer experiment conditions which would be extremely difficult, if not impossible, to achieve in the laboratory. Finally, the Chester, Gann, Gallagher, and Grimison chapter demonstrates that the computer mystique is being displaced by a hard-headed appraisal by chemists of what computer enhancements are possible with existing technology. Through the addition of a peripheral device (designed to do floating-point arithmetic very rapidly) to the campus largescale data-processing machine, it was possible to improve the cost effectiveness of the total syste simulation of matter. The growing sophistication of matter modeling as a result of advances in computer size, speed, and cost effectiveness may well have as its greatest impact, however, the enhancement of communication among those interested in any particular physical system. Illinois Institute of Technology Chicago, Illinois October 13, 1978

PETER LYKOS

xi In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1 Molecular Dynamics Simulations of Liquids with Ionic Interactions K. HEINZINGER, W. O. RIEDE, L. SCHAEFER, and GY. I. SZÁSZ Max-Planck-Institut fuer Chemie (Otto Hahn-Institut), Mainz, Germany

Molecular dynamics (MD) simulations of liquids with ionic interaction have so far been performed for molten salts and aqueous electrolyte solutions. The characteristic problem for this kind of simulation are the far ranging Coulombic forces. The f i r s t preliminary MD calculations for molten salts have been reported by Woodcock in 1971 (1). The large amount of work published in the meantime has been reviewed by Sangster and Dixon (2). In the case of aqueous electrolyte solutions only preliminary results for various alkali halide solutions have been published so far (3). The more advanced state of the art for the molten salts leads to a concentration of the effort on the improvement of the algorithm for the integration of the equation of motion necessary to calculate dynamical properties with still higher accuracy. One example where very high accuracy is needed is the calculation of the isotope effect on the diffusion coefficients. In the section on molten salts below one way to improve the algorithm is d i s cussed and the progress is checked on the mass dependence of the diffusion coefficients in a KCl melt. Results with a certain degree of r e l i a b i l i t y from MD simulations of aqueous solutions reported up to now are restricted to structural properties of such solutions. In the section on aqueous solutions below very preliminary velocity autocorrelation functions are calculated from an improved simulation of a 0.55 molal NaCl solution. The problem connected with the stability of the system and the different cut-off parameters for ion-ion, ionwater and water-water interactions are discussed. Necessary steps in order to achieve quantitative results for various dynamical properties of aqueous electrolyte solutions are considered. Molten Salts The Hamilton differential equation system can be solved 0-8412-0463-2/78/47-086-001$07.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

2

COMPUTER MODELING OF MATTER

numerically for a N particle system with a simple "leap-frog" algorithm as used by Verlet (4) or by a predictor-corrector a l gorithm. The most important criterion for the choice of the a l gorithm is the numerical s t a b i l i t y . The final decision follows from necessary accuracy with which for example the energy is preserved, which in turn is determined by the desired accuracy of the transport properties, say self diffusion coefficient. In the simulated system of molten KC1 i t has been investigated how the total energy, the total momentum, and the velocity autocorrelation functions varies with different algorithms. The MD calculation for the KCl-system was carried out at the melting point (1043K). The basic cube contains N=2-108 particles and the cube length S is than calculated to be 20.6A. For the pair potential the Born-Mayer-Huggins potential ip(r) = + er"

1

+ bexp(-Br) + Cr" + Dr' 6

(1)

8

is used with the parameter in Table I. Table I Parameters for the Born-Mayer-Huggins Potential b

+

K

+

-

K

+

- CI"

CI"'-

cr

D

C

10~ erg

A-1

1991.67768

2.967

1224.32459

2.967

-

4107.95500

2.967

- 145.5

12

K

B

10" erg A 12

-

6

10" erg A8 12

24.3

-

24.0

48.0

-

73.0

- 250.0

Three different algorithms were investigated. In the f i r s t version (I) the Coulombic energies and forces were evaluated with the use of the erfc part of the Ewald method (6) only. In the other two versions (II and III) the f u l l Ewald method was employed. In a l l versions the separation parameter T), the number of nearest neighbours n~ , and the maximum value of the reciprocal lattice vectors h] (not occuring in version I) were chosen to be: i\ = 0.175,

n^ = 6,

h

1

= 8.

The algorithm which was used in version I is a so called "leapfrog" algorithm: Ii(n+1) = r.(n-1) + 2At v.(n-1) + (At /m.) JF.(n) 2

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(2)

1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions

3

v.(n) = v^n-1) +[v.(n-1) - v.(n-2)] + (A^/m^F.fn) Here r.-j( )> v.i( )> n

n

Lj( )

a n c l

n

a

r

e

t

h

e

- ^(n-1)]

(3)

position, velocity and force

of particle i at time t = nAt respectively and m-j are the masses -14 of the ions. The time step length was chosen to be 0.5-10 sec. The behavior of the total energy for version I is shown in F i gure 1 where (AE/E) =6 - 10-3. A change from the "leap-frog" algorithm to a predictorcorrector algorithm has been made in version II. For the position vectors one can write: r?(n+1) = r?(n-1) + 2 At v.(n-1) + r?(n+1) = r?(n-1) + 2 At v.(n-1) + (At /4m.)[ F.(n+1) + 4F.(n) + 3F.(n-1)] 2

(5)

Here the indices p and c indicate predicted and corrected values respectively. The formula for the velocities has been changed slightly compared to the formula (3): ^ ( n ) = v.(n-1) + (At/2m.)[F (n) + F.(n-1)] i

(6)

With this version II, two simulations with different time steps were carried out starting from the same configuration (0.5 10"1^s and 0.25 10~14s) over a time interval of 1.2 ps. The total energies for both runs are shown in Figure 1 . The value (AE/E)Q ^ has decreased compared to the one of version I by one order of magnitude. Moreover AE/E decreased further by a factor 0.5 when the time step was shortened to 0.25.10"14 . In version III the positions and velocities have been treated with the same predictor-corrector algorithm. This means that F-j ( t ) , the derivative of the forces with respect to time, appears in the formula for the velocities. s

rP(n+1) = r?(n-1) + 2 At v?(n-1) + (2At /3 ) [2F (n) + F.(n-1)] 2

mi

i

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(7)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

J

1 01

Etotal

1 •

1 05

.

.

3

.

.

, 1 psec

.

.

•

3

Figure 1. Total energy of version I ('upper curve) in units of 10 J/mol and of version II flower curve) with a energy scale enlarged by a factor of 4 A. The length marked by (jimm) in the upper scale corresponds to the interval of the lower scale. The time scale starts after an equilibration run over 7.2 psec from a roughly equilibrated starting configuration.

627 6

633-

1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions

r?(n+1) = r<:(n-])

+ 2 At v9(n-1)

5

+ ( A t ^ . ) [F.(n+1) + 6F.(n) + 5F.(n-1)] 2

+ (At /24m.) [3^.(0-1) - ZF.(n) - F.(n+l]

(8)

3

and

for the velocities: vj(n)

= v?(n-2) + 2 At/m. F.(n-2) + (2At /3 ) [2F.(n-1) + F.(n-2)] 2

vj(n)

(9)

mi

= v^n-1) + (At/2m.)[ F.(n) + F.(n-1)] + (At /12m.) [F.(n-1) - F.(n)]

(10)

2

It has to be emphasized that there is a basic difference between the algorithms of version II and III. In version II the veloc i t i e s were not handled in the same way as the positions by a predictor-corrector algorithm. This is less time consuming per time step and might be sufficient for a given accuracy in the total energy for example. The values of (AE/E)Q 5 of Version III decreased by two orders of magnitude compared to version I. When the time step was shortened by half, ( A E / E ) Q . 2 5 decreased one order of magnitude more. Finally A E / E decreased by three orders of magnitudes compared with A E / E of version I. An compilation of a l l A E / E is given in Tab.II according to the different versions and time steps. Table

II

Values of AE/E according to the different versions version

I

At-10" s

0.5

AE/E

6 10

1 4

II 0 . 25

0. 5 -

3

6

10

III

- 4

3 10-

0 . 25

0. 5

3 10"

4

5

2 10-

6

The fluctuation of the kinetic energy is about +2500 J/mol corresponding to a fluctuation of temperature of about + 100K. Compared with the total energy i t follows (AE. . / E ) 2 5 " that the fluctuations of the kinetic <*nergy'are three orders of magnitude larger than the fluctuations of the total energy and hence the fluctuations of the kinetic energy are almost compensated by the corresponding fluctuations of the potential energy. In order to keep the computing time small for each time step, care has been taken to calculate the right hand side (r.h.s) of the differential equation system only once per time step. This n

0

= 4 # 1 0

3 ,

T

h

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

i

s

s

h

o

w

s

6

COMPUTER MODELING OF MATTER

means that the predicted values rP should not be too different from the corrected values r?. Witft version III

n?-iii<

£

for each component (x,y,z), each time step, and a l l particles i where e = 10~ A. Moreover for about 70% of the particlese = 10~?A could have been chosen. From eq. (7) and (8) follows 6

|Fi(ri) - F.(n+1) -[F.(n+1) - F.(n)] -(At/4)[3F.(n-1) - 2F(n) - JF.(n+1 ]\<

fem./At

2

(12)

This inequality can be estimated i f one replaces the differentiation by the corresponding ratio of the differences which yields: fAF.(n)

- AF.(n-1)|< 4em./At

(13)

with F-j(n) - Fi(n-1) = AF-j(n). Eq.(13) shows that the change of the force change within the given time step determines the ratio e/At^ for constant mass. The desired accuracy of the particle positions requires an adequate time step At and determines the accuracy of the forces. Eq.(13) would be meaningless i f the errors of the forces were as large as the changes of the force change, because the correction added to the predicted value of jy would be of the order of the errors of the forces. However,e has to be small enough because one wants to calculate the forces with the predicted values of rj only. An analogous equation to eq.(11) is given for the velocities with ane'which is different from e . |vP

-v;|<e-

(14)

Fore'a value of 20 cm/s has been found which corresponds to a maximum change in the positions of 5-10-6A. This shows that the positions of the particles will be slightly changed by taking the corrected velocities rather than the predicted values in eq.(8). Moreovere corresponds to a maximal temperature difference AT which can be estimated by 1

AT<(m/3k)e'[ e'+ 2 (3kT/m) ] 1/2

(15)

Here m is the average mass, k Boltzmann's constant, and T the temperature at the melting point. The value of the r.h.s. of eq. (15) is 0.5 K which is 5 °/oo of the temperature fluctuations of about + 100 K. This means that the temperature fluctuations are mainly determined by the change of the velocities in time rather than by the corrections of the velocities.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1. HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions

7

The absolute value of the total momentum |IP| of the system is shown in Figure 2 . It decreased by almost one order of magnitude between version I and II. Almost no change occured in |PJ comparing version II with III. Therefore only |PJ of version III has been plotted in Figure 2 . However, a decrease by a factor of 4 shoved up when the time step was reduced. Additionally, |PJ was reduced by a factor of about 3 when the i n i t i a l configuration also included the differentiation of the forces. This total momentum curve is marked with a capital A in Figure 2 . A more precise analysis of the total momentum make i t reasonable to assume t (16) 0

Hence the total force F(t) does not vanish completely for each time step, the total momentum is the sum of the deviations of the total force from zero, rather than the total momentum of the system in a physical sense librated i n i t i a l configuration. The extrema of the total momentum components in Figure 3 correspond to the zeros of the total force components at the same time points. Moreover the forces of a l l particles have been separated into the different parts of the pair potentials, eq.(1), and have been summed up separately,

2

f( )

2 (> 8

(17) 4—1 ; 1 Here the indices c, ex, 6, and 8 indicate to which part of the pair potential the forces belong. The calculation has shown that a l l terms of eq.(17) except the coulomb term are smaller than 3-1(r dyn/mol for each component while the coulomb term is seven orders of magnitude larger, 3-10^ dyn/mol. Consequently, the deviation of the total force from zero is determined by the sum of coulombic forces. The velocity autocorrelation functions and from them the self diffusion coefficients have been calculated from a l l versions for the normal masses as well as for the mass of the anion reduced by 26%. They are shown together in Figure 4 . The velocity autocorrelation functions in version I do not behave qualitatively correct. If the anion mass decreases, the f i r s t zero of the anion correlation function should shift to smaller time, while the zero of the unchanged cation correlation function should slightly shift to larger time. The qualitatively correct behaviour of the velocity autocorrelation functions results from version II and III. The behaviour of the total energy, kinetic energy, the i n stantaneous value of P (t) = (2E| -j -i|>)/3V , and the velocity autocorrelation functions of the system have been investigated with a change of the cut-off radius in the interaction which belongs to the pair potential part 1/r6. Here is the v i r i a l of the system calculated by N (18) «|» = - £ H j F ( r i j ) 11 +

c

n

6

+

F

m

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978. 3

4

y

Figure 2. Absolute total momentum in units of 10 cm g/sec mol on the left-hand scale while the right-hand side relatively gives the absolute momentum to the averaged momentum per partide. The upper curve for version I, the lower curves for version III, with a scale enlarged by a factor of 4 A.

>

^

00

° o

5

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions

Figure 3. Total momentum P in units of 10* cm g/sec mol total force F in units of 10 dyn/mol for the x,y,z components (solid, dashed, and dotted line, respectively) y

12

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

Figure 4. Velocity autocorrelation functions from version 7, //, and III for K* ( Cl ( Cl (•— Cl (+ +). Vertical lines allow zeros of the correlation functions to be compared. 39

35

31

27

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

MD Simulations of Liquids with Ionic Interactions 11

HEINZINGER ET AL.

No cut-off radii have been changed for the other interactions. In calculation (I) a l l 215 particles in the environment cube (the cube with edge of length S and particle i at i t s center) of part i c l e i have been taken into account for the calculation of the force acting on particle i and the energy. In a second run (II), which was started from the same configuration,all the particles have been taken into account within the sphere which circumscribes the environment cube of particle i . The total energy decreased as expected by about 0.02%) and stayed constant within (AE/E)Q.25 - ' " the whole run of 1440 time steps. As no iteration happened in either of the calculations I and II, eq.(13) s t i l l holds especially for the beginning of the runs. This result shows that the deviations have been of the order of the allowed uncertainties of the forces. The kinetic energy and P (t) differ more and more from each other as a function of time as is shown in Figure 5 . However, no qualitative changes P (t) have occured. Althoug in the forces F^' are very small compared to the forces F-j themselves, a significant change appeared in the kinetic energy. Thus a small uncertainty in the forces at each time step which is not purely random or is not cancelled by summation, might produce a trend for example in the kinetic energy. Furthermore a slightly different starting configuration causes a quantitatively but not qualitatively different kinetic energy. This fact allows the choise of the starting velocity vector (3N-dimensional) independently of the time sequence of calculation of these vectors with regard to the averaging procedure for the autocorrelation functions. This is discussed in ref.(7) in more d e t a i l . However, the cut-off radius does influence the velocity autocorrelation functions as shown in Figure 6 and the corresponding time integrations which lead to the diffusion coefficient as shown in F i gure 7 . The differences between the velocity autocorrelation functions of runs I and II are larger than the root-mean-square error of the mean for part C (see curve I C and II C in F i gure 6 . It is obvious that this error could lead to an underestimation of the errors for the velocity autocorrelation functions, as is discussed in more detail in ref. (8). Moreover, when comparing the velocity autocorrelation functions from Figure 6 with the ones from Figure 4 for the lighter anion mass, i t seems to be necessary to have a higher accuracy for larger times (say round about 0.5 psec), at least as high as in version III, in order to get a significant difference in the self diffusion coefficients. In order to reduce these errors, i t seems to be better to increase the number of particles per cube rather than to increase the number of time steps. A larger number of particles brings about another improvement besides a better s t a t i s t i c s . The necessity of introducing periodic boundary conditions (discussed in detail in many places, e.g.(1,2,8,12)) has some influence on the =

1

6

1 0

6

o

v

e

r

+

+

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

T

3.6 p

Figure 5. Temperature as function of time steps in the upper figure (solid line, calculation I; dots, calculation II). P* as function of time steps in the lower figure (heavy solid line, calculation I; light solid line, calculation II).

2000 -

2000

£000 -

900

1000

1100

1200

1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions 13

Figure 6. Velocity autocorrelation functions plus and minus the root-meansquare error of the mean averaged over 60 starting velocity vectors for each part A, B, and C (Figure 5) of calculations I and II respectively (solid line, K\ dashed line, Cl) 39

35

0 6 psec

Figure 7. Integration of the velocity autocorrelation functions (Figure 6) leading to the self-diffusion coefficient for large times (approximately 0.5 psec). The nomenclature is equivalent to that in Figure 6.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

14

system too, as can be seen for example from the different values of the v i r i a l obtained, depending whether one calculates

*

=

" I "ij i j F(r

)

°

r

7J The difference will decrease for the non Coulombic interactions i f the number of particles is increased,since the number of particles whose interaction sphere contains periodic images will become smaller than the number of particles whose interaction sphere contains no periodic images. This means that the cut-off radius for the non Coulombic interaction has to be sufficiently large, so that the absolute value of the particle - furthest particle force is small enough compared to the absolute value of the force acting on a particle; but the cut-off radius also has to be small compared t In the above discussion the Coulombic interaction was l e f t out. With the long range Coulombic forces, i t is possible that the periodic boundary conditions being used here will lead to behaviour not present in a real physical system, when a f i n i t e size cube is used. However, i t might be possible to assess the effect of the boundary conditions on the model by increasing the number of particles per cube. At present no practical alternative more physical model has been demonstrated. (A model which may eliminate the periodicity problems has been proposed by Friedman (20). To the best knowledge of the authors this proposal has not been incorporated yet into a computer algorithm). Aqueous Solutions The rigid ST2 point charge model used for the simulation of pure water by S t i l l i n g e r and Rahman (9) with its rotational degrees of freedom is employed for the MD" calculations of aqueous solutions. It requires a different kind of algorithm than the one used for molten salts. A stabilized predictor - corrector formalism is employed which has been developed by Gear (10) for the solution of differential equations of a l l orders. This algorithm is based on a method by Adam and improved for an easy change of the time step length by Nordsieck (IJ). Pure water has been s i mulated with the same algorithm (12). The method is based on the knowledge of the f i r s t N time derivatives of the variables y i ( t ) at a given time. For the intergration of the equation of motion

i" the predicted values are calculated through a Taylor series expansion, p.,- = 2 for the centre of mass coordinates x- and p-j = 1 for the Euler angles (or quaternion parameters) a-j and angular p

n

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

MD Simulations of Liquids with Ionic Interactions 15

HEINZINGER ET AL.

velocities.

If ( At)

a. =

dy

J

J

( 23)

and a are (N+1)-dimensional vectors with the components a , a ^ , . . , a , then N

4:lt= i A

2 4

is the predicted value, where A is the Pascal Triangle A The predictor - corrector formulae have the form (m+1) _ (m) ^-t+At " t + A t

X

G

a

(

<>

t

k l

, (m) ^-t+At)

=(u(

2

b

)

X = (X-pAo, •••»^N+I) ^ y that the integration remains stable nummerically after n time steps. \ depends on the order of the integratio chosen for the translational and a seventh order for the rotational motion. The higher order for the rotational motion is necessary because of the faster change of the Euler angles compared with the translational movement, a consequence of the small moments of inertia of the water molecule. It follows from the order of the integration procedure that the local truncation error for the centre of mass coordinates is a function of A t and for the Euler angles and angular velocities a one of At&. The situation for the simulation of aqueous solutions is demonstrated with a 10,000 time step run of a 0.55 molal NaCl solution. The basic periodic box contained 200 water molecules, 2 sodium and 2 chloride ions. The ST2 point charge model is employed and the alkali and halide ions are treated as point charges residing at the centre of Lennard - Jones spheres. Each of the six effective pair potentials consists of a LJ term T

1

S

c

h

o

s

e

n

i

n

s

u

c

a

w a

6

V t j ( r ) =4

E

i

j

{ ( c^/r)

1

-

2

( c^./r) }

(26)

6

where i and j refer either to ions or water molecules and a Coulomb term, different for water-water, ion-water and ion-ion interactions, as given by ^ C(r.d

1 1 t

d

1 2

.

...) =S (r).q .

d ' 2 •-> + 1 (-w)(-1) (-2)

V

d

+

2

w w

=(

+

>

2

0=1

H )

J ^

e

I

a

+

B

/ d

a

a (-a)

d

+

V (r) = + 2 / r ++

e

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

B

16

COMPUTER MODELING OF MATTER

The switching function, S ^ r ) has been introduced to reduce unr e a l i s t i c Coulomb forces between very close water molecules ( 1 2 ) . e is the elementary charge and q = 0.23 e is the charge in the ST2 water model, d and r denote distances between point charges and LJ centres, respectively. The choice of a and 6, odd for positive and even for negative charge yields the correct sign. The LJ parameters for water-water interaction are taken from the ST2 model, and for cation-cation from the isoelectronic noble gases (Y3). The anion-anion parameters are derived from the cation ones on the basis of the Pauling radii as shown in detail in a previous paper (14). The LJ parameters between different part i c l e s are calculated by application of Kong's combination rules (J5J. A l l e and o are given in Table III. Table

III

Lennard - Jones parameter in a NaCl solution is H0

Na

3.10

2.92

4.02

54.84

30.83

2.73

3.87

5937

28.32

-

27.87

2

H0 2

Na cr

+

52.61

-

+

CI"

4.86

The classical equations of motion are integrated in time steps of At = 1.09-10-16 (22).In order to take account of the different strengths of the various interactions and to keep the computer time in reasonable limits, different cut off parameters are chosen for ion-ion, ion-water and water-water interactions. The Ewald summation (6), which gives the correct Coulombic energy for an unlimited number of image ions, is used for ion-ion interactions. As the ions residing in the f i r s t and second neighbour image boxes are included in the direct calculation of the so called error function part, a suitable choice of the separation parameter allows to neglect the Fourier part of the Ewald sum. The ion-water interaction is cut off at 9.1 A, half the sidelength of the basic periodic box. The water-water interaction is cut off at 7.1 A which means that only f i r s t and second nearest neighbour molecules are included in the calculation. The structural properties of the NaCl solution calculated from this simulation are not significantly different from preceding ones. The characteristic distances an heights of the ionwater and water-water radial pair correlation function remain a l most unchanged whether a temperature control mechanism is built s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

HEINZINGER ET AL.

MD Simulations of Liquids with Ionic Interactions 17

in or not. The radial pair correlation functions from the simulation of a 2.2 molal NaCl solution (200 water molecules and 8 ions of each kind) are compared in detail with X-ray investigations in a previous paper (14). The problem connected with the determination of hydration numbers have also been discussed previously (16). In addition i t has been concluded from the MD s i mulations of alkali halide solutions that in the f i r s t hydration shells a lone pair orbital of the water molecules is directed towards the cation and a linear hydrogen bond is formed with the anion (1?). This result is also in agreement with conclusions from experimental and other theoretical investigations. The real aim of MD simulations i s , of course, the calculation of the dynamical properties of the liquid. The velocity autocorrelation function have been calculated from this 10.000 time step run. In a l l cases the functions have been averaged over 360 starting vectors. The s t a t i s t i c a l quality is much better, of course, in the case of average extends over 20 of the ions. In Figure 8 the angular (solid line) and the translational (dashed line) velocity autocorrelation functions of the water molecules are shown. The correlation time for the angular velocity is as expected much shorter than for the translational velocity. In Figure 9 the velocity autocorrelation functions for the cation (solid line) and the anion (dashed line) are drawn. If these autocorrelation functions are integrated, in spite of a l l s t a t i s t i c a l uncertainties i t is found that the corresponding diffusion coefficients are of the same order of magnitude as is known or estimated from experimental investigations. Therefore i t is j u s t i f i e d to continue the investigation along this line. It is obvious that much longer simulation times are required to make quantitative statements about the dynamical properties. But not only an increase in simulation time is necessary, also the simulation i t s e l f needs further improvement. The system simulated corresponds to a microcanonical ensemble and consequently i t should have a constant total energy after an equilibration time of suitable length. In Figure 10 the total energy (solid line) and the potential energy (dashed line) for the system are shown over the 10.000 time steps run. I can be seen that the increase of the total energy is equally distributed between kinetic and potential energy. The unavoidable cumulative errors due to the algorithm, which were discussed above in detail in the case of the molten s a l t s , are not the main source of energy increase in the simulation of aqueous solutions. Responsible for the problems here are mainly the cut-off r a d i i . Figure 11 shows the variations of the total energy of the basic box E during a simulation run of 270 & t, which is essentially caused by the cut-off radii R and Ri in the evaluation of the potential energy E t ( t ) . Let us consider the expression: w

p o

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

18

COMPUTER MODELING OF MATTER

Figure 9. Normalized velocity autocorrelation functions qp(t) for cations (solid) and anions (dashed in a 0.55m NaCl solution)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1. HEINZINGER ET AL.

M D Simulations of Liquids with Ionic Interactions 19

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

20

COMPUTER MODELING OF MATTER

-232 AE

-233

50

100

150

200

250 At

Figure 11. Variation (cut-off jumps) of the total energy of the basic box in units of 10 erg for 200 water molecules and eight Ntf CZ. The time step is 2.18 • 16 sec and AE/E = 3 • 10 . 12

16

4

s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

E

MD Simulations of Liquids with Ionic Interactions 21

HEINZINGER ET AL.

pot

( t +

A t

>- pot E

•

tV (r (t^t)-V (r (t))}

( t ) =

i

1 j

1 j

i j

^(r^tt^t))-

1 j

^(r^t))

Z

(28)

where the summation extends over i and j such that I

r^.JtJ^R and ^ .(t+At)
II

r..(t+

III

r.jtt+AtJ^R^r.jtt)

At)^R
.

The last two terms are due to the fact that a particle i located within the spherical environment with radius R of a particle j at a certain time may leav step, or vice versa. Usin R one can estimate the last term of eq. (28) as:

As the interactions depend on the orientations of the particles, averaged potentials V , V , and V are considered here, n (t) l f t l

WW

—

W>

w

W

(n (t) and n_(t) ) is the number of water molecules which leave one of the cut-off environments of central water molecules (ions) during a time step. Furthermore, using the number density of water, one gets n (t)

^

R

n (t)

&

R.

w

+

2

(3kT/m )

2

(

w

w

3 k T

/

m

+

)

1/2

V 2

At

(l^ - 1) = n,

At

(N ) = n +

2

(30)

with N the number of water molecules and N-j the number of ions in the basic periodic box. m and m- are the masses of water molecules and ions, respectively, Table IV shows quantities from our simulations. w

w

n

Table IV Quantities relevant to energy jumps due to the cut-off radii for a 2.2 molal NH C1 solution. 4

w-w cut-off

w-i cut-off

R

7.1 A

9.1 A

|V(R)|/E

2 • 10"

n

1.5

4

7 -10" n

+

4

= 1/10, n_ = 1/14

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

22

COMPUTER MODELING OF MATTER

1/ri2 is the average number of time steps after which a jump occurs. The fluctuations and the upward trend in both the translational and rotational averaged kinetic energy of the water molecules are shown in Figure 12 in Kelvin. The much faster f l u c tuations of the rotational energy are obvious. As expected a fast exchange of energy between rotational and translational degrees of freedom occurs. It is demonstrated in Figure 13 that the distribution of rotational energy is Maxwellian. This is also true for the other subsystems. In Figure 14 the trend in the total kinetic energy is clearly recognizable together with the much stronger fluctuations in the case of the ions where only two of each kind are in the basic periodic box. The lighter sodium f l u c tuates significantly faster than the chloride ion. It is known from previous simulations at higher NaCl concentration that the upward trend in the total energy increases with concentration. As the ion-io rather high accuracy (Ewal the energy increase has to be attributed to the cut off radius of the ion-water interaction. The upward trend in energy is also a hint that during the stepwise integration of the equation of motion the stabilizing effect of the integration algorithm gets partially lost because of cumulative errors. It is shown now that inaccuracies in the force calculations may lead to a loss in the stabilizing effect of the corrector. Let us consider center of mass coordinates (com) x, and use the abreviations of eq.(28). The time step is choosen in such a way, that only one iteration is needed in the predictor-corrector formalism. Therefore one has X2(t) = (At)2/2F ( t ) , and the forces F (t) are calculated with predicted com coordinates and Euler angles. Starting from an i n i t i a l value x ( t ) , the algorithm yields for x ( t + A t ) : n-1 P , x ft+nAt)=x ft) + EI [x (t+mAt)+F(t+mAt) AtV2J 0

0

1

At2 r "° + ^ - [ A A F ( t + n A t ) + A _ AF(t+(n-1)At)+ . . . m

n

+A AF(t+At)] +a 1

n

n

1

x (t)+b 3

n

x (t)+c 4

n

x (t). g

(31)

The coefficients A-j and a , b , c can be calculated from the stabilization coefficients X j . If the difference of the forces between two time steps n

n

n

AF(t+mAt) = F(t+mAt)-F(t+(m-1) A t )

(32)

is not much larger than the inaccuracies in the evaluation of the forces. F , a stabilizing effect cannot be expected from the Ai ,an,b ,cn. For a given system, F(t+mAt) depends on the magnitude of the time step and can therefore easily be changed. Ferr then e r r

n

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

HEiNziNGER ET AL.

MO Simulations of Liquids with Ionic Interactions 2 3

CO

Ο

"δ

to to ο

to

C to ο

H2

^

3

II .2 to

5

"•*» c Ο

3

^

II Î3 i

1 ο

<Ν

to 3

Ε

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1

°' "I

1

100

1

1

1

1

500

1

1

" ° " ^ ° H ^ 1

•

1000

1

K

Figure 13. Comparison of the normalized rotational kinetic energy distribution of the water molecules with a Maxwellian one (dots)

dN dT

to

M D3

>

p

o w r

§

I

4^

1. HEINZINCER ET AL.

1

MD Simulations of Liquids with Ionic Interactions

1 0 25

r 0.5

1 075

psec

Figure 14. Kinetic energy of the total system ( )> the cations ( anions ( )in Kelvin as a function of simulation time

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1

1

—

j, and

26

COMPUTER MODELING OF MATTER

depends both on the accuracy of the programmed force routines and, as F(t) is calculated from predicted values, on the accuracy of the predicted com coordinates and Euler angles. Simulations of aqueous solutions are characterized by the fact that the changes in the coordinates in one time step depend strongly on the type of motion. The choise of At for such a s i mulation is essentially determined by the very fast changes in the Euler angle coordinates, and At is thus in principle too small for the integration of the translational motion. A At of 1.09-10-16 sec leads to a contribution from the corrector of lArl
Go

= - a s i n p cosy + (3siny

GD

= dsin p siny + PCOSY ,

x

a)

z

$

(33)

=d cosp + y ,

(for the definition of the Euler angles compare (19) ) become singular i f sinP = 0. Two solutions of this problem are possible. In the f i r s t case, the Euler angles are redefined, when sin0 approaches 0 (12). This method is used for the simulation of the a l c a l i halide solutions. The second method is to use the so called quaternion parameters (qp) (18). These are defined in terms of the Euler angles as X = cos p/2 cos (a +y )/2 t| = -sinp/2 sin (a - y)/2 g

= -sin (3/2 cos (a - y )/2

£

=

(34)

cos p/2 sin (a + y )/2

with the auxiliary condition: ? ? ? ? X + T I + £ + £ = 1 and lead to singularity-free equations instead of eq. (33). This type of parametrisation is connected with the spin-1/2 representation of the rotation group D(a,p,y), as can be seen from the expressions containing half Euler angles. One easily verifies

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1.

MD Simulations of Liquids with Ionic Interactions 27

HEINZINGER ET AL.

feosp/2 exp(i(a+y)/2)

sinp/2 exp(-i(a-y)/2)\

i-sinp/2 exp(i(a-y)/2)

cosp/2 exp(-i(a+\)/2)

D(ct,p, ) = Y

/X+

i£

- Z+

U

in

X-

+

(36)

K

The equations to be used instead of (33) are cu y (XI z l

£

X/

The s t a b i l i t y of the energy was not essentially improved after building in the quaternioni that increasing the time step At by a factor of 2 leads to an increased upward trend in energy i f the Euler angle substitution procedure is used, while the energy remains the same for the s i mulation using the singularity-free procedure. The latter check was preformed for an ammonium cloride solution ( 2 0 0 H2O 8 NH4+ and 8C1"). In this case, the NH4+ molecule has a rigid tetrahedral four-point-charge distribution and also takes part in the rotational motion. Conclusions Some problems of numerical s t a b i l i t i e s in molecular dynamic calculations of liquids with ionic interactions have been discussed. Mainly a sufficient high accuracy seems to be necessary for the intergration and force routine in order to get the kinetic properties of the system from long simulation runs. In the case of aqueous solutions the main source of inaccuracy at this stage is the cut-off radius for ion-water interactions. It could be solved by the introduction of a method s i milar to the Ewald summation. An improvement in the intergration of the equations of motion seems to be possible by the choice of different time steps for translational and rotational motion. The pair potentials employed seem to describe aqueous solutions appropriately. It might be necessary at a later stage to introduce changes here too. Then the problem of the boundary conditions mentioned below must also be investigated. A different approach to the simulation of aquous solutions might be based on the central force model employed by Sti1 linger and Rahman (2J_) for pure water. In simulations of ionic melts periodic boundary conditions appear which lead for the Coulomb interactions to the use of the Ewald method. It seems necessary to investigate the influence of

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

28

COMPUTER MODELING OF MATTER

the periodic boundary conditions and the Ewald method on the transport properties by increasing the number of particles per cube. This will also have the effect of better s t a t i s t i c s for quantities like the pressure where the increased number of part i c l e s is more important than very long runs in time. Acknowledgement Financial support by Deutsche Forschungsgemeinschaft is gratef u l l y acknowledged. Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22.

Woodcock, L.V., Chem. Phys. Letters (1971), 10, 257. Sangster, M.J.L. and Dixon, M., Advances in Physics (1976), 25, 247. Heinzinger, K. an 31a, 463. Verlet, L., Phys. Rev. (1967), 159, 98. Tosi, M.P. and Fumi, F.G., J. Phys. Chem. Solids (1964) 25, 31. Ewald, P.P., Ann. Physik (1921), 64, 253. Schäfer, L. and Klemm, A., Z. Naturforsch. (1976), 31a, 1068. Schäfer, L., Z. Naturforsch., to be publisched. S t i l l i n g e r , F.H. and Rahman, A., J. Chem. Phys. (1974), 60, 1545. Gear, C.W., ANL Report No. ANL-7126 (1966). Nordsieck, A., Math, of Computation (1962), 16, 77-80, 22. Rahman, A. and S t i l l i n g e r , F.H., J. Chem. Phys. (1971), 55, 3336. Hogervørst, W., Physica (1971), 51, 59, 77. Pálinkás, G., Riede, W.O. and Heinzinger, K., Z. Naturforsch. (1977), 329, 1137. Kong, C.L., J. Chem. Phys. (1973), 59, 2464. Bopp, P., Heinzinger, K. and Jancso, G., Z. Naturforsch. (1977), 32a, 620. Heinzinger, K., Z. Naturforsch. (1976), 31a, 1073. Evans, D.J., Mol. Phys. (1977), 34, 317. Goldstein, H., Classical Mechanics, Addison-Wesley, Cambridge, Mass. (1953). Friedman, H.L., Mol. Phys. (1975), 29, 1533. S t i l l i n g e r , F.H. and Rahman, A., J. Chem. Phys. (1978), 68, 666. Vogel, P.C. and Heinzinger, K., Z. Naturforsch. (1975), 30a, 789.

RECEIVED August 15,

1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

2 Monte Carlo Simulation of Water C. S. PANGALI, M. RAO, and B. J. BERNE Columbia University, New York, NY 10027

The usual Metropoli Carlo(1,2) procedur whe applied to the study of water a bottlenecks in configuration space. In the Monte Carlo (M.C.) method different configurations of the system are sampled according to the Boltzmann distribution: where Rj is a six component vector in the case of water. Metropolis et al. devised a scheme according to which a new configuration R' = (R'1,...R'N) is generated from an old configuration R = (R1,...RN) by sampling a transition probability T(R'|R), and R' is accepted with probability

and rejected with probability q = 1 - P. Normally the configuration is changed by selecting a particle j at random, or cyclinically, and displacing it from Rj to R'j where dR.j = Rj - Rj is sampled uniformly between —— and + ——. The transition probability for an atomic fluid is therefore given by

where D defines a domain about each point R. in which the moves are to be sampled. Such a method when suitably modified for polyatomic fluids does not converge rapidly, particularly at low temperatures. In fact the M.C. method gives higher potential energies and lower values for the specific heat than the 0-8412-0463-2/78/47-086-029$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

30

COMPUTER MODELING OF MATTER

molecular dynamics method f o r the same state(3_). I f we —8 v (R ) T(R!|R_.) « , i t i s seen from Eq. (2) t h a t p = 1.

take

1

e

v(R')

i s not known, we

expand i t around C

T (R'. IR.)

e

-BV^ ~R.

R, and

, OR. e D otnerwise

v(R)«6R. ~ ~j

3

=

Since

so

where C i s a n o r m a l i z a t i o n constant and V

(4)

i s the g r a d i e n t

R

J operator with r e s p e c t to p o s i t i o n s and angles. I t i s easy to see t h a t t h i s procedure s a t i s f i e s the c o n d i t i o n o f d e t a i l balance: e"

B v ( R )

T(R'|R) P(R'|R) =

e

"

B

v

(

R

,

)

T(R|R«) P(R|R')

(5)

We adopted the Barker-Watt M e t r o p o l i s M.C. walk. The change i n o r i e n t a t i o n i s accomplished by s e l e c t i n g the x, y or z a x i s a t random and performing a r o t a t i o n through an angle 69 about the chosen a x i s . The angle A0 A9 69 i s u n i f o r m l y d i s t r i b u t e d on (—— , ~^~~) • Our new f o r c e - b i a s 0

0

method samples t r a n s l a t i o n a l displacements 6x., 6y. and 6z. according to the components of the f o r c e F. acting^on the p a r t i c l e j , while 69. w i l l be sampled s u b j e c t to the component of the torque along t h e a x i s s e l e c t e d f o r the r o t a t i o n . 3

3

Results and D i s c u s s i o n S t a r t i n g from a l a t t i c e c o n f i g u r a t i o n with 216 H O molecules i n t e r a c t i n g with an ST-2 potential(5) a t T = 283 K, the standard M e t r o p o l i s method generated a walk i n which the p o t e n t i a l energy p e r ^ p a r t i c l e f l u c t u a t e d around -130.5e ± 0.2e (e = 5.31 x 10 e r g ) . A l t o g e t h e r n e a r l y 500,000 moves were attempted with t h i s method. S t a r t i n g with one of the " e q u i l i b r a t e d " s t a t e s from t h i s method, our f o r c e b i a s scheme generated a q u i t e d i f f e r e n t walk. The r e s u l t s are summarized i n Table I. Not o n l y does the f o r c e b i a s method g i v e a s t a t e with a lower mean p o t e n t i a l energy, but the s t a t e i s r e p r o d u c i b l e . S t a r t i n g with a con­ f i g u r a t i o n generated by M.D. a t 283°K with V = -135.5e, the standard scheme generates a walk i n which = -135.0c ± 0.2e whereas the forge b i a s method y i e l d s = -134.0e ± 0.4e. At T = 391 K the two M.C. methods and the molecular dynamics method a l l give the same e q u i l i b r i u m s t a t e with s t a t i s t i c a l l y i n d i s t i n g u i s h a b l e p o t e n t i a l energies. In the molecular dynamics method the p a i r f o r c e i s s e t equal to zero when the p a i r d i s t a n c e exceeds a c e r t a i n d i s t a n c e r . Thus the t r a j e c t o r i e s correspond to a s h i f t e d p o t e n t i a l Q

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

2.

Monte Carlo Simulation of Water

PANGALI ET AL.

31

(J)(r) - ^ ^ Q ) where <M Q) depends on the o r i e n t a t i o n . The M.C. method, on the other hand deals with a truncated but u n s h i f t e d potential. Thus the disagreement noted i n Table I between M.D. and f o r c e b i a s M.C. may be a s c r i b e d t o t h i s d i f f e r e n c e between the two methods. To check out t h i s a s s e r t i o n we studied a d r o p l e t o f water c o n t a i n i n g 27 p a r t i c l e s a t T = 283 K, i n t e r a c t i n g with i n f i n i t e range ST-2 p o t e n t i a l . M.D. and the f o r c e b i a s M.C. give i d e n t i c a l r e s u l t s . In c o n c l u s i o n we note t h a t Monte C a r l o s t u d i e s on h i g h l y s t r u c t u r e d f l u i d s can be very misleading. At low temperatures t h i s procedure leads t o c o n f i g u r a t i o n a l b o t t l e n e c k s . Although the f o r c e b i a s method i s p r e f e r a b l e t o the standard M e t r o p o l i s scheme, i t i s important t h a t any sampling method be thoroughly t e s t e d t o ensure that a unique e q u i l i b r i u m s t a t e i s reached. R

T = 283°K



a

Metropolis Force b i a s Molecular Dynamics

Acceptance Ratio 0.43 0.65

-130.5±0. 2 -133.5±0. 4 -137.4

a) In u n i t s o f e, e = 5.31 x 10~ -1 -1 b) In u n i t s o f c a l mol K .

-19.0 -23.0 23.8

•15 erg.

Literature Cited 1.

Metropolis, A. H., and

N., M e t r o p o l i s , A. W., Rosenbluth, M. N., T e l l e r , Teller E. J . Chem. Phys. (1953) 21, 1087.

2.

V a l l e a u , J . P. and Whittington, S. G., A Guide t o Monte C a r l o for Statistical Mechanics: 1. Highways, in:"Statistical Mechanics, Part A: E q u i l i b r i u m Techniques"(Berne, B. J. e d . ) , Plenum Press, New York(1977).

3.

Ladd, A. J. C., Mol. Phys. (1977) 33., 1039.

4.

Watts, R. O., Mol. Phys. (1974) 28, 1069.

5.

S t i l l i n g e r , F. H., and Rahman, A., J. Chem. Phys. (1974) 60, 1545.

RECEIVED August 15, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

3 Determination of the Mean Force of Two Noble Gas Atoms Dissolved in Water C. S. PANGALI, M. RAO, and B. J. BERNE Department of Chemistry, Columbia University, New York, NY 10027

To b e t t e r understan non-polar molecules d i s s o l v e average f o r c e between tw (Lennard-Jone Spheres) solved i n ST2 water as a f u n c t i o n o f the separation r between the two spheres. Each xenon atom i n t e r a c t s with every water molecule through a Lennard-Jones ( 1 2 - 6 ) p o t e n t i a l with parameters (o = 3.43Å, ε = 1 7 0 . 1 ° K ) . The water molecules i n t e r a c t with each other through the f a m i l i a r ST2 p o t e n t i a l (1). Molecular dynamics simula­ t i o n s were performed f o r the two Xe atoms f i x e d a t a separation r f o r s e v e r a l values o f r ranging from 5.35Å t o 9 . 1 8 Å . In each of the f i v e simulations there were 214 water molecules at a den­ sity o f 1 g cm-3. The temperature f l u c t u a t e d around 298°K. The time step f o r the numerical i n t e g r a t i o n was Δt = 4.2 x 10-16 sec f o r e q u i l i b r a t i o n and h a l f that f o r the final r u n . The initial phase p o i n t f o r a separation o f 9-18A was generated from e q u i l i ­ b r a t e d pure water by choosing two water molecules already a t the desired separation, f i x i n g t h e i r p o s i t i o n s , turning o f f t h e i r coulomb i n t e r a c t i o n s and a d j u s t i n g t h e i r p o t e n t i a l parameters t o the Xe-H 0 parameters. Further e q u i l i b r a t i o n i s c a r r i e d out f o r 2 , 0 0 0 At. This was followed f o r 3 , 0 0 0 At during which the ave­ rage f o r c e on each Xe atom was computed. A f t e r t h i s run a t a p a i r separation o f 9.18A, the p a i r d i s t a n c e was reduced and the system was e q u i l i b r a t e d . In t h i s way, runs were produced f o r the other p a i r d i s t a n c e s s t u d i e d . Two Xe atoms l a b e l e d a and b, d i s s o l v e d i n water w i l l be d i s t r i b u t e d according t o the p a i r c o r r e l a t i o n f u n c t i o n , g ( r ) , aD g ( r ) = exp - I w ( r ) ] a b

a b

where v ^ ( r ) i s the p o t e n t i a l o f mean f o r c e . The average f o r c e along the l i n e j o i n i n g the Xe atoms i s given by a

- V v f r ) = < F >r ab ~a

~b

0-8412-0463-2/78/47-086-032$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

3.

PANGALI ET AL.

Mean Force of Noble Gas Atoms in Water

33

<

where F > and are the average f o r c e s on t h e two Xe atoms determined i n t h e molecular dynamics experiment. This can "be expressed as. 3v f r ) a

- - s r ^

=

+

<

f S o 1 v

^ >

where F ( r ) i s the c o n t r i b u t i o n due t o the d i r e c t Lennard-Jones i n t e r a c t i o n and < F ( r ) > i s the f o r c e between the atoms induced by t h e solvent molecules. L J

s o l v

In t a b l e 1, we r e p o r t how the solvent induced force < F > v a r i e s with r . The e r r o r s are estimated from the magnitude o f the f o r c e components p e r p e n d i c u l a r t o the l i n e j o i n i n g a and b . S t r i c t l y speaking, these should be zero. C l e a r l y much longer runs are necessary t o improve the accuracy o f the work reported here. We found i t d i f f i c u l cause at the l a r g e s t s e p a r a t i o s t a n t i a l , making t h e t a s k o f i n t e g r a t i n g impossible. S O l v

The i n t e r e s t i n g f e a t u r e o f our r e s u l t s i s t h e o s c i l l a t i o n i n ; t h i s i n d i c a t e s t h a t when t h e two Xe atoms are a d i s t a n c e g r e a t e r than 5.5A a p a r t , a water molecule i s l i k e l y t o take a p o s i t i o n between them. T h i s i s c o n s i s t e n t with the two Xe atoms r e s i d i n g i n two cages with a monolayer o f water between them. A f t e r completion o f t h i s work we l e a r n e d that A. Geiger, et a l (h ) s t u d i e d the h y d r a t i o n o f two Lennard-Jones s o l u t e p a r t i c l e s . T h e i r r e s u l t s show a q u a l i t a t i v e s i m i l a r i t y t o ours, but they do not give a q u a n t i t a t i v e r e s u l t f o r t h e p o t e n t i a l o f mean f o r c e . Our work supports the theory o f P r a t t and Chandler (2) on the hydrophobic e f f e c t . Previous Monte Carlo work (3) along s i m i l a r l i n e s d i s p l a y e d no o s c i l l a t i o n i n , although i n t h a t study the authors explored values o f r smaller than 7. Table

F

L J

o

2e

O

w

and e

-7.55 -U.30 -1.19 -0.58 -0.29

w


s o l v

O

>a W

2e w

5..35 6,.07 7..U6 8..30 9..18

I

w

19.89 ± 5 . 2 -20,20 ±k.9 3.23 ±6.2 35.88 ±10.2 33.7^ ± 5 . 8

w

2e w

12.3 * -5.2 -2^.50 ±k.9 2.05 ±6.2 36.h6 ±10.2 3^.03 ± 5 . 8 1

r e f e r t o the parameters f o r the ST2 p o t e n t i a l .

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

34

COMPUTER MODELING OF MATTER

Literature 1.

F. H.

Stillinger,

Cited

and A. Rahman, J. Chem. Phys. 6 0 , 1545

(1974) 2.

L. P r a t t and D. Chandler, J. Chem. Phys. 67, 3683

3.

V.G. Dashevsky and G.N. Sarkisov, Mol. Phys. 27, 1271

4.

A. Geiger, A. Rahman, and F.H.

RECEIVED August 15,

Stillinger,

(1977)

(Preprint)

1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(1974)

4 Applying the Polarization Model to the Hydrated Lithium Cation CARL W. DAVID Department of Chemistry, University of Connecticut, Storrs, CT 06268

Electrostatic model chemistry. The Polarization Model has a genealogy which extends back to the work of Born and Heisenberg (1) and Rittner (2) i n which simple electrostatics was applied to ionic compounds. Modern examples of such models include the recent work of E t t e r s , Danilowicz, and Dugan (3), and, i n Inorganic Chemistry, the work of Guido and Gigli (4). Since H e a v i s i d e ' s theorem precludes purely electrostatic stability of ionic compounds, a repulsive potential of some sort must be included to stabilize an isolated ionic molecule in the gas phase. The freedom to introduce such an ad hoc repulsive potential suggested to various people that even wider lattitude might be had i n order to extend electrostatic models to include covalently bonded compounds. Our concern here i s with water and its interactions, and we specialize to that substance now. Early models of water, attempting to account for polymeric properties of the species using quasi-electrostatic schemes, placed imaginary negative charges i n the vicinity of the oxygen atom. The l a t e s t such venture is, already, almost a classic. The Ben-Naim S t i l l i n g e r ( B N S ) p o t e n t i a l (5) was used i n d i s c u s s i n g liquid water up to and i n c l u d i n g d i r e c t numerical i n t e g r a t i o n of Newton's equations of motion for water molecules (6). The p o t e n t i a l was an outstanding success. Several other p o t e n t i a l s for the water-water i n t e r a c t i o n have r e c e n t l y appeared, as has a r e v i s i o n of the BNS p o t e n t i a l known as ST2 (7, 8, 9, 10). Tn addition, several model p o t e n t i a l s have appeared which incorporate p o l a r i z a t i o n effects (11, 12). Each of these p o t e n t i a l s seems quite capable of being 0-8412-0463-2/78/47-086-035$06.75/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

36

COMPUTER MODELING OF MATTER

used t o compute p r o p e r t i e s o f m a c r o s c o p i c samples o f water. These e l e c t r o s t a t i c p o t e n t i a l s f o r water a l l s u f f e r a f a t a l d e f e c t , as they a p p l y t o f r o z e n w a t e r not i c e - b u t water m o l e c u l e s w i t h fixed (i.e., f r o z e n ) geometry. I n an a t t e m p t t o e n l a r g e t h e c l a s s of water p o t e n t i a l s , Lemberg and S t i l l i n g e r f!3) introduced a central force model w h i c h a l l o w e d " u n f r e e z i n g " t h e water m o l e c u l e . Here ?id hoc atom p a i r i n t e r a c t i o n s are introduced which, i n t o t o , bind the water m o l e c u l e t o g e t h e r at i t s eouilihrium geometry while simultaneously allowing water molecules t o i n t e r a c t p r o p e r l y among t h e m s e l v e s . T h i s model was t e s t e d i n a r e c e n t m o l e c u l a r dynamics s i m u l a t i o n (14) q u i t e s u c c e s s f u l l y . The c e n t r a l f o r c e model does n o t s u f f e r from t h e f r o z e n geometry defec models. The m o l e c u l dissociates. However, i t i s i n t e r n a l l y u n a f f e c t e d by e x t e r n a l e l e c t r o s t a t i c f i e l d s , and t h e r e f o r e does n o t f u l l y mimic r e a l w a t e r . An e l e c t r o s t a t i c model o f t h e water monomer, which included polarizability i n some r e a s o n a b l e manner m i g h t a l l o w f o r complete s i m u l a t i o n o f l i o u i d water and o f i o n i c solutions. The s t a t i c d i e l e c t r i c properties of this ubiquitous s o l v e n t would, in a good model, be a u t o m a t i c a l l y c o r r e c t , both on t h e microscopic and on t h e m a c r o s c o p i c l e v e l . The thermodynamic p r o p e r t i e s p r e d i c t e d by such a model w o u l d , e x c e p t f o r m i n o r c a l i b r a t i o n e r r o r s , be c l o s e to q u a n t i t a t i v e . Having no s e r i o u s doubts about m i m i c k i n g water a t l i q u i d d e n s i t i e s , one can p r o c e e d to u s i n g more s o p h i s t i c a t e d models o f t h e w a t e r molecule. The r a i s o n d ' e t r e need no l o n g e r bo j u s t i f i c a t i o n o f t h e methodology, and one can s e t t l e down t o p r e d i c t i o n s and c o r r e l a t i o n s confident that the e r r o r s i n t h e p r e d i c t i o n s a r e i n a c c u r a c i e s i n t h e model. The P o l a r i z a t i o n

Model

The P o l a r i z a t i o n Model r e g a r d s a water m o l e c u l e as a c l u s t e r o f 2 p r o t o n s ( w h i c h r e p e l each o t h e r ) and an o x i d e i o n ( w h i c h i s p o l a r i z a b l e ) . The b i n d i n g o f the o x i d e i o n t o the p r o t o n s i s e f f e c t e d through an empirical potential function. The d i p o l e moment o f the m o l e c u l e i s adjusted t h r o u g h two special e l e c t r o s t a t i c m o d i f i e r f u n c t i o n s , K ( r ) and L ( r ) . The p r o t o n s g e n e r a t e an e l e c t r i c f i e l d at the o x i d e a n i o n w h i c h , i n r e a l e l e c t r o s t a t i c s , would have

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

Polarization Model and the Hydrated Lithium Cation

DAVID

the

37

form 3

F - e r /

(1

r

where e i s t h e charge on t h e p r o t o n . Fince the electrons o f r e a l water prevent t h e proton from actually s e t t i n g up such a f i e l d , and s i n c e t h e p r o t o n s would s e t up no f i e l d i f r=0, i t is r e a s o n a b l e t o i n v e n t a s h i e l d i n g f u n c t i o n which w i ] l account f o r these two f a c t s , i . e . , modify electrostatics so t h a t i t i s applicable i n this a r t i f i c i a l model. The f u n c t i o n c h o s e n , K f r ) has l i m i t s such t h a t , a t l a r g e r , K ( r ) goes t o z e r o , thereby returning t h e e l e c t r o s t a t i c s which i s e x p e c t e d . However, a t s m a l l r , t h e f u n c t i o n K f r ) a c t s to c u t o f f t h e e f f e c f i e l d so t h a t a t r the f i e l d c o m p l e t e l y . The f o r m a l d e f i n i t i o n o the function i s : 3

( 1 - K(r) )

G - e r/ r

(2

G i v e n t h e e f f e c t i v e e l e c t r i c f i e l d (G) a t the o x i d e i o n due t o t h e p r o t o n s , one has, i f the oxide i o n i s p o l a r i z a b l e , t h e i n d u c t i o n o f a d i p o l e moment in the anion. The magnitude o f t h i s induced moment is 3

p

= a e r/ r

( 1 - K(r))

(3

Associated with this i n d u c e d d i p o l e moment on t h e o x i d e i o n ( w h i c h opposes t h e n u c l e a r d i p o l e moment) i s a p o l a r i z a t i o n energy = (1/2)

U

u . E

(4

pol Fere, we choose to again modify normal electrostatics. Instead o f using the true e l e c t r i c field i n E q u a t i o n 4, and i n s t e a d o f usir><* t h e m o d i f i e d e l e c t r i c f i e l d o f E q u a t i o n 2, we i n v e n t y e t another e m p i r i c a l f u n c t i o n , L ( r ) which m o d i f i e s t h e electric field i n t h e energy term t o account a g a i n f o r t h e i n t e r v e n i n g e l e c t r o n s and f o r t h e symmetry o f the problem a t r = 0 :

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

38

COMPUTER

MODELING

OF

MATTER

3

E so

f

«

e

r / r

( 1 - J,(r) )

(5

that 3

U

-

(e/2)

y-r / r

( 1 - I (r) )

pol (6

The e m p i r i c a l f u n c t i o n s K ( r ) and L ( r ) are s e t by f o r c i n g v a r i o u s o b s e r v e d p r o p e r t i e s o f water to agree with the model. The f u n c t i o n s are d i s p l a y e d i n F i g u r e 1 i n the form o f l - K ( r ) and l-I.(r) Their c a l i b r a t i o n i s c a r e f u l l y d i s c u s s e d i n r e f e r e n c e (15"). The t o t a l energy o f a water m o l e c u l e i s g i v e n by: E

«

2 e /r

• V HI!

(r 0-H

) + V 1

(r 0-I

) + U T

(7 no]

2

where V i s a p o t e n t i a l energy f u n c t i o n which l o o k s , at large r , to be C o u l o n b i c , and w h i c h mimics the e f f e c t o f the c h e m i c a l bond i n h o l d i n g the p r o t o n s to the o x i d e anion. The c u r v a t u r e and depth this potential have been chosen to g i v e good agreement w i t h e x p e r i m e n t a l f a c t s about w a t e r . This potential function i s also displayed in Figure 1 . E q u a t i o n 7 r e p r e s e n t s the i n t e r n a l energy a water m o l e c u l e . In an assembly o f water m o l e c u l e s , one requires o n l y an Oxygen-Oxygen interaction f u n c t i o n ( p r o t o n s i n t e r a c t among themselves v i a pure Coulombic f o r c e s i n t h i s model) and one may p r o c e e d to a p p l y the model to any system c o n s i s t i n g of p r o t o n s and o x i d e i o n s a l o n e . This f u n c t i o n i s also discussed i n reference (15). The model, parameterized as renin red, successfully accounts for the "trans-linear" s t r u c t u r e o f the gas-phase water d i n e r , which i s h e l d together by an e l e m e n t a r y hydrogen bond. It s u c c e s s f u l l y p r e d i c t s the o r d e r o f s t a b i l i t y o f the eas-phase t r i m e r s o f w a t e r ( i . e . , agrees w i t h the e x i s t i n g quantum m e c h a n i c a l o r d e r i n g ) , and appears q u i t e c a p a b l e o f s e r v i n g as a v i a b l e p o t e n t i a l eneray f u n c t i o n f o r use i n Monte C a r l o and m o l e c u l a r dynamic simulations. The d e t a i l s o f the a p p l i c a t i o n o f t h i s model to v a r i o u s i o n s o f the form:

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Figure 1. The calibrated functions required for the Polarization Model

4x

> In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

40

COMPUTER

MODELING

OF

MATTER

II 0 m n has

a l s o been g i v e n i n r e f e r e n c e ( 1 5 ) . Ve e x p e c t an a l m o s t one-to-one correspondence between the h y d r a t e d p r o t o n and t h e h y d r a t e d L i t h i u m cation. Lithium Cation-Water P o t e n t i a l C a l i b r a t i o n f

S i n c e t h e e x a c t p o t e n t i a l energy o a Lithium c a t i o n i n t h e f i e l d o f a water m o l e c u l e i s unknown, i t i s c o n v e n i e n t t o assume t h e a p p r o x i m a t e v a l i d i t y of the Kistenmacher, Popkie, and C I e m e n t i ( K P C ) "simple" p o t e n t i a l energy f u n c t i o n f ! 6 ) . This f u n c t i o n was i n v e n t e o b t a i n e d energy h y p e r - s u r f a c fixed nuclear g e o m e t r y ) w a t e r m o l e c u l e s i n the f i e l d of a Lithium c a t i o n . Although i t i s true that a " b e t t e r " f i t t i n g f u n c t i o n t o t h e H a r t r e e - F o c k energy s u r f a c e i s known and r e p o r t e d , t h e s i m p l e f o r m , which shows a minimum energy a t about -35 k'cal/mol at a L i t h i u m - O x y g e n d i s t a n c e o f 1.9 A, i s adeouate f o r our purposes. The ground rules for parameterizing the P o l a r i z a t i o n Model f o r t h e L i t h i u m c a t i o n (and any other species) i s s u b j e c t t o no f i r m theoretical f o r m u l a t i o n . A p r o t o c o l suggested by S t i l l i n g e r (17) a s s i g n s K ( r ) and L ( r ) f u n c t i o n s to s p e c i e s on t F e b a s i s o f how they appear i n t h e f o r m a l e q u a t i o n s o^ the P o l a r i z a t i o n M o d e l . The q u e s t i o n i s moot i n t h e cases under i n v e s t i g a t i o n h e r e , as t h e Lithium-Oxygen and L i t h i u m - H y d r o g e n d i s t a n c e s w i l l a l l be so l a r g e under normal c i r c u m s t a n c e s t h a t we can s a f e l y d e c l a r e b o t h K and L t o be z e r o . T h i s i s eouivalent to allowing their electrostatic i n t e r a c t i o n s t o be "classical". P r a g m a t i c a l l y , t h i s means employing K and L as i s from t h e z e r o - o r d e r Polarization odel, as t h e s e former K and L f u n c t i o n s a r e a l l zero a t these l a r g e d i s t a n c e s . The v a l u e o f a f o r the L i t h i u m c a t i o n , r e q u i r e d i n t h e Model i s chosen (18) t o have the v a l u e M

3 .03 A The o t h e r i n t e r a c t i o n s which must be c o n s i d e r e d a r e t h e L i t h i u m - p r o t o n i n t e r a c t i o n , and t h e L i t h i u m Oxygen i n t e r a c t i o n . The former i s purely repulsive,

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

Polarization Model and the Hydrated Lithium Cation

DAVID

41

and t h e r e i s l i t t l e sense i n being more r e p u l s i v e than the d e f a u l t Coulomb r e p u l s i o n . V e t h e r e f o r e s e t out the ground r u l e t h a t we w i l l not t i n k e r w i t h r e p u l s i v e p o t e n t i a l s . T h i s l e a v e s the Lithium-Oxygen i n t e r a c t i o n as the o n l y i n t e r a c t i o n term w i t h w h i c h we can adjust the i n t e r a c t i o n p o t e n t i a l energy between t h e c a t i o n and water so t h a t the total potential energy mimics (even i f weakly) the t h e o r e t i c a l l y derived p o t e n t i a l functions. A l t h o u g h t h e r e i s no i n t r i n s i c m e r i t t o u s i n g t h e s m a l l e s t number o f parameters in fitting the p o t e n t i a l e n e r g y , i n t h i s c a s e , we have chosen t o f i t the KPC p o t e n t i a l t o t h e P o l a r i z a t i o n o d e l by e m p l o y i n g one e x p o n e n t i a l term i n the Lithium-Oxygen i n t e r a c t i o n , which a l s o , o f c o u r s e , includes the Coulomb a t t r a c t i o n o f the two o p p o s i t e l y charged ions. This f i t t i n g f u n c t i o n r

M

-3.58 15820 e

r rkcal/mol)

r e s u l t s i n a compromise f i t t o the KPC p o t e n t i a l i n the c o n f i g u r a t i o n i n w h i c h the two p r o t o n s are as f a r as p o s s i b l e from the L i t h i u m i o n , and the L i t h i u m Oxygen a x i s i s a symmetry a x i s . In F i g u r e 2 t h e computed p o t e n t i a l energy function f o r t h e KPC " s i m p l e " model and the P o l a r i z a t i o n Model a r e d i s p l a y e d f o r the symmetric planar c o n f i g u r a t i o n shown. The f i t i s a d e o u a t c . Figure 3 shows t h e KPC p o t e n t i a l versus the P o l a r i z a t i o n Model p o t e n t i a l i n the reversed planar c o n f i g u r a t i o n i n d i c a t e d . The compromise n a t u r e o f the " f i t " i s apparent. Much t o our s u p r i s e , r e l a x a t i o n o f the s t r u c t u r e from the planar KPC c o n f i g u r a t i o n r e s u l t s in a distinct increase i n t h e bond strength of the complex, and, more i m p o r t a n t l y , r e s u l t s i n a d i s t i n c t tetrahedr.al c o n f i g u r a t i o n . The a b s o l u t e minimum energy o f the L i t h i u m water cation complex i s -1072.28 k c a l / m o l , b e i n a 39. kcal/mol more s t a b l e than isolated water. This minimum i s a l s o t o be compared t o the minimum energy shown i n F i g u r e 2 w h i c h was -34.39 k c a l / m o l . AV e x p e r i m e n t a l l y f o r the p r o c e s s +

Li is

-34.0

kcal/mol

(19).

+

+ H 0 2

T

L i (I 0^1 2

R e l a x a t i o n o^ the w a t e r has

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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R _
2,0

0

3

X

)

4

0

5

.

0

Figure 2. The Kistenmacher Popkie Clementi potential function for the lithium cation vs. water and the total Polarization Model potential for the same geometric configuration, as functions of the lithium-oxygen distance. ( ) Polarization Model, a = .03 A . ( ) K.P.C. potential. 3

20

3 R

L i - 0

. (

0 A

4

0

5

0

)

Figure 3. The same potential functions as in Figure 2 hut with the configuration reversed so that the protons of the water molecule are pointing at the lithium cation. ( ) Polarization Model, a = .03 A . ( J K.P.C. potential. 3

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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resulted i n structure stabilization o f about 5 kcal/mol. Whether o r not t h i s s t a b i l i z a t i o n i s r e n l o r not i s open t o q u e s t i o n . We have chosen t o al^ow t h i s extra 5 kcal/mol s t a b i l i z a t i o n t o remain i n t h e p a r a m e t e r i z a t i o n , hoping t h a t a better quantumc h e m i c a l c o m p u t a t i o n based on t h e s e s u g g e s t i o n s v/511 v a l i d a t e our a s s u m p t i o n . (One s h o u l d n o t e t h a t o^ a l l the i o n s measured by K e b a r l e and D z i d i c , this mono-hydrate i s the o n l y one whose An v a l u e was o b t a i n e d by e x t r a p o l a t i o n . T h e r e f o r e , more than w i t h any o f t h e o t h e r i o n s , there i s a n o n - t r i b a l u n c e r t a i n t y about the r e p o r t e d A J v a l u e o f --34. A kcal/mol.) Both 0-H bond l e n g t n s i n water have expanded i n the complex, g r o w i n g from 0.95P4 A t o 1.005 A. F u r t h e r , the I -0-I a n g l e has i n c r e a s e d t o 106.05 d e g r e e s . The L i - 0 d i s t a n c e has shrunk from the " f i t " v a l u e o f 1. A A. Lewis (20) t e t r a h e d r a l i t Y

J

0+ =

360. -

Li-O-II

T

1

Li-O-I!

2

Jf V -°-F 1 2

we f i n d f o r the t o t a l l y r e l a x e d complex a 9+ v a l u e o f 17.4 d e g r e e s . I n t h i s measure, 9+= 0 c o r r e s p o n d s t o p l a n a r i t y , and 6+ « 31.5 degrees c o r r e s p o n d s t o a tetrahedral configuration(where the lone-nair e l e c t r o n s on a r e a l oxygen atom would be p o i n t i n g d i r e c t l y a t the c e n t r a l c a t i o n ) Friedman and Lewis quote an e x p e r i m e n t a l v a l u e o f 11 degrees f o r the 9+ value observed i n s o l i d L i SO :2(V O) 2 4 2 They f u r t h e r r e p o r t a bond l e n g t h f o r the l i t h i u m Oxygen "bond" o f 1.91 A. From the p o i n t o f v i e w o* the P o l a r i z a t i o n M o d e l , t h e r e e x i s t s no such thin<» as lone-pair electrons. The n o n - p l a n a r structure n e v e r t h e l e s s o c c u r s u n l e s s one assumes t h a t t h e Lithium proton i n t e r a c t i o n should be made more r e p u l s i v e than C o u l o m b i c . I t appears as i f Vaslow (21) was the f i r s t t o s u g g e s t t h a t the d i p o l e o f a water m o l e c u l e might not point directly at a central cation under c i r c u m s t a n c e s such as t h o s e b e i n g c o n s i d e r e d h e r e . T h i s p o i n t o f view d i d n o t r e c e i v e theoretical c o r r o b o r a t i o n from E l i e z e r and K r i n d e l f 22^ nor does i t r e c e i v e such from Quantum M e c h a n i c a l c o m p u t a t i o n s which may be found i n t h e c u r r e n t l i t e r a t u r e (23 -31). Spears (32) a l s o f i n d s the symmetric p l a n a r

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complex to be most s t a b l e ( 3 3 ) . The l a c k o f t h e o r e t i c a T c o n f i r m a t i o n does not preclude the possibility that the non-nlanar s t r u c t u r e i s , i n f a c t , the more s t a b l e one. In t h e i r c l a s s i c s e t o f c o m p u t a t i o n s on i o n s of the s e r i e s +/II 0 n m Newton and Fhrenson (34) o b t a i n e d a minimim energy c o n f i g u r a t i o n f o r the Hydronium i o n which was p l a n a r , contrary to expectations. A more expensive c o m p u t a t i o n by Kollman and Pender (35") was r e q u i r e d to o b t a i n p y r a m i d a l d i s t o r t i o n ( 3 6 ) . "The s u b s t i t u t i o n of Lithium c a t i o n f o r proton i n the hydronium i o n i s not an i s o e l e c t r o n i a n a l o g y need not h o l d p a i n f u l l y c l e a r t h a t the i o n s h o u l d not be p l a n a r . Since high a c c u r a c y Quantum M e c h a n i c a l c o m p u t a t i o n s are q u i t e e x p e n s i v e , one can perhaps u n d e r s t a n d why " f r o z e n " geometry water m o l e c u l e s have been used i n these L i t h i u m - w a t e r complex i o n c o m p u t a t i o n s , and t h e r e f o r e why p l a n a r , t r i g o n a l s t r u c t u r e s have always been found to be most f a v o r e d . There e x i s t s a modicum o f e x p e r i m e n t a l e v i d e n c e suggesting that i n concentrated L i T and IiCl solutions the Li-H distances are smaller than e x p e c t e d f o r p l a n a r s t r u c t u r e s . O e i g e r and P e r t z (37) f i n d t h a t i n 6m L i l s o l u t i o n s the L i - J d i s t a n c e i s between 2.7 A and 2.46 A, which, assuming an L i - 0 d i s t a n c e o f 2.08 A r e s u l t s i n a canted water l i ^ a n d , w h i l e Narton, Vaslow and Levy (38) find a Li-D d i s t a n c e o f 2.43 A i n 7m L i C l soTution. Of c o u r s e , such c o n c e n t r a t e d s o l u t i o n s are not e q u i v a l e n t to our "gaseous" i o n s , and the q u e s t i o n of the s t r u c t u r e o f the i o n s under d i s c u s s i o n here remains open. 1

Complexes o f the L i t h i u m C a t i o n w i t h Water. The d i - h y d r a t e o f the L i t h i u m c a t i o n i s f o u n d , using the P o l a r i z a t i o n Model as an i n v e s t i g a t i v e tool, to have two w e l l d e f i n e d minimum erergy conformations. If one imagines the starting g e o m e t r i e s from w h i c h r e l a x a t i o n l e a d s to energy m i n i m i z a t i o n , t h e r e a r e two s i m p l e ones which come to mind. With a w a t e r on each s i d e o f the c e n t r a l i o n , one i s the t o t a l l y p l a n a r complex w h i l e the second i s the n o n - p l a n a r s t r u c t u r e i n w h i c h one o f the water ligands i s canted a t 90 degrees r e l a t i v e to the

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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o t h e r . The d i f f e r e n c e i n energy which we compute f o r the r e s u l t a n t minimum energy c o n f i g u r a t i o n s s t a r t i n g from each o f these two i n i t i a l c o n f i g u r a t i o n s i s o n l y 0.1 kcal/mol, c l e a r l y an i n s i g n i f i c a n t difference when one r e a l i z e s t h a t t h e t o t a l energy isa d i f f e r e n c e between l a r g e numbers. One o f t h e s e c o n f o r m a t i o n s i s shown i n F i g u r e 4 . In both c a s e s , the water m o l e c u l e s d i s t o r t a p p r e c i a b l y from their planar starting c o n f i g u r a t i o n s ( w i t h r e s p e c t t o the central c a t i o n ) . The energy o f h y d r a t i o n f o r t h e monohydrated c a t i o n ( t o form the d i h y d r a t e d c a t i o n ) i s -30.54 k c a l / m o l , ( e x p e r i m e n t a l AH v a l u e = -25.8 kcal/mol) averaging over t h e two low energy conformers. The 9+ v a l u e obtained for the s t a g g e r e d ( p l a n a r ) conformers was 17.32(18,84) degrees f o r 01, and 16.73(17.80) degrees f o r 02. Both water m o l e c u l e s have H-Oand a l l 0-I d i s t a n c e d i s t a n c e s have i n c r e a s e d s l i g h t l y from t h e i r values i n the mono-hydrate, t o 1.91 A. The h i g h e r h y d r a t e s o f L i t h i u m c a t i o n are a l l o f i n t e r e s t , p r i m a r i l y because the e n t h a l p y o f h y d r a t i o n f o r these i o n s has been measured. T h e r e f o r e we have u n d e r t a k e n to f i n d low energy c o n f o r m a t i o n s o f these h i g h e r h y d r a t e s , a l t h o u g h we a r e w e l l aware o f the f a c t t h a t the a b s o l u t e minimum energy c o n f i g u r a t i o n s are not n e c e s s a r i l y the ones we a r r i v e a t through s t r a i g h t - f o r w a r d m i n i m i z a t i o n t e c h n i q u e s . Tn f a c t , i t is clear from our e x p e r i e n c e w i t h h y d r a t e d p r o t o n s and h y d r a t e d h y d r o x i d e i o n s t h a t the h y p e r - s u r f a c e s i n q u e s t i o n a r e pock-marked w i t h h i l l s and v a l l e y s , and t h a t the a b s o l u t e g l o b a l minimum c o n f o r m a t i o n may not be a c c e s s i b l e from a l l starting configurations due t o a l g o r i t h m i c c o n s t r a i n t s . With t h i s c a v e a t i n mind, we d e s c r i b e the h i g h e r h y d r a t e s o f the L i t h i u m cation. The b e s t , i . e . , the l o w e s t energy, conformation we can f i n d f o r the tri-hydrated cation isa m o d e r a t e l y p l a n a r one which i s non-symmetric. The c o m p e t i t i o n i n s i d e the f i r s t c o o r d i n a t i o n sphere f o r hydrogen-bonding i s becoming a p p a r e n t , a n d , as a r e s u l t , one o f the water m o l e c u l e s i s pushed out from the c e n t r a l c a t i o n so t h a t i t can a c c e p t a hydrogen bond from one o f the n e a r e r l i g a n d s . To be s p e c i f i c , one Oxygen i s l o c a t e d 3.78 A from the L i t h i u m c a t i o n , w h i l e the o t h e r two a r e l o c a t e d a t 1.86 A and 1.93 A. As c a n be seen i n F i g u r e 5, t h e n e a r e s t water m o l e c u l e ( t o the c e n t r a l c a t i o n ) , i s d o n a t i n g a p r o t o n to the most d i s t a n t water m o l e c u l e . The r e s u l t a n t HO-F a n g l e f o r t h i s donor m o l e c u l e i s s m a l l e r than T

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normal, 103.9 d e g r e e s . The a c c e p t o r w a t e r has a normal H-O-H bond a n g l e o f 104.75 d e g r e e s . The donor m o l e c u l e s O-II d i s t a n c e s a r e 0.99 and 1.05 A, w i t h the l o n g O-F bond b e i n g the d o n a t i n g one to the a c c e p t o r ligand. The 9+ v a l u e f o r the donor l i g a n d i s 24.95 degrees(the highest value yet obtained), w h i l e the a c c e p t o r l i g a n d has a 9+ v a l u e o f o n l y 12.79 degrees. The lonely third l i g a n d has a 9+ v a l u e o f 16.81 degrees and i s d e p r e s s i n g l y normal i n a l l r e s p e c t s . The energy o f t h i s t r i - h y d r a t e d L i t h i u m c a t i o n i s found to be -3186.6 kcal/mol, resulting in a h y d r a t i o n energy f o r the di-hydrate of -18.0 k c a l / m o l , compared t o the e x p e r i m e n t a l A V v a l u e o f -20.7 k c a l / m o l . I t i s not a t a l l c l e a r what meaning t o a s c r i b e to a s t r u c t u r e such as t h a t j u s t p r e s e n t e d f o r the t r i hydrated Lithium c a t i o n no evidence, i n an whether or not this structure i s correct, or reasonable, or whatever. The l a s t hydrated complex which w i l l be considered in this study i s the auadri-hydrated Lithium cation. A f t e r extensive(and expensive), but not exhaustive, geometric s e a r c h i n g , we find a d e e p e s t l o c a l minimum energy c o n f i g u r a t i o n f o r t h i s complex w i t h energy o f -4223.5 k c a l / m o l . This c o r r e s p o n d s t o an energy o f h y d r a t i o n f o r the t r i hydrated cation o f -4 kcal/mol. This i s an u n r e a s o n a b l y low v a l u e r e l a t i v e t o the e x p e r i m e n t a l v a l u e o f AH = -16.4 k c a l / m o l r e p o r t e d by K e b a r l e and Dzidic. The geometry o f the complex i s not p a r t i c u l a r l y encouraging e i t h e r . As can be seen i n F i g u r e 6, the l o w e s t energy c o n f i g u r a t i o n i s not tetrahedral, as expected or a n t i c i p a t e d . Rather, t h r e e o f the w a t e r s are t r i g o n a l l y d i s p o s e d around the L i t h i u m c a t i o n , w i t h the f o u r t h water o c c u p y i n g a position r o u g h l y d e s c r i b e d as b e i n g i n the second coordinate s h e l l . The bond l e n g t h s f o r the v a r i o u s Li-0 are, i n t h e i r order as d e p i c t e d on F i g u r e 6, 1.91, 3.91, 2.54, and 2.04 Angstrom. The w a t e r closest t o the L i t h i u m c a t i o n i s the o n l y one p a r t i c i p a t i n g i n a hydrogen bond, (as d o n o r ) , and i t is bonding t o the o n l y water i n the second c o o r d i n a t i o n sphere. T h i s p a r t i c u l a r donor water has a v e r y s m a l l H-01-F a n g l e , 99.7 d e g r e e s . A l l the o t h e r F-0(n)-H a n g l e s are " n o r m a l " , i . e . , 103.9, 104.9 and 106.1. The 9 a l u e s a r e o f some i n t e r e s t i n t h i s c a s e , as the Ol 9+= 51.21 degrees. This i s , of course, exceptionaTTy h i g h . One must however b e a r i n mind + V

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Figure 4. One of the two minimumenergy conformations found for the dihydrated lithium cation

Figure 5. A low-energy structure (possibly a minimum) found for the trihydrated lithium cation

Figure 6. A low-energy structure (possibly a minimum) found for the fourhydrated lithium cation. The symmetric structure is of higher energy than this one, although there is no guarantee that the reported structure is a global minimum.

American Chemical Society Library

1155 16th St., N.W. Washington, D.C. 20036 In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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t h a t 01 i s p a r t i c i p a t i n g i n two bonds, one to the Lithium cation, and the o t h e r to a n o t h e r water m o l e c u l e ( i n the second c o o r d i n a t i o n s p h e r e ) . The 9+ v a l u e f o r 03 i s c l o s e r to n o r m a l , 11.5 d e g r e e s . The 9+ v a l u e f o r 04 i s 20.0 d e g r e e s . There i s l i t t l e doubt t h a t t h i s s t r u c t u r e i s not observable. We have c a r e f u l l y s e a r c h e d the n u c l e a r configuration space i n the vicinity of the symmetrically disposed t e t r a h e d r a l l y coordinated water complex, h o p i n g to f i n d a minimum energy c o n f i g u r a t i o n w h i c h might b e t t e r c o r r e s p o n d to the i n t u i t i v e l y a p p e a l i n g p i c t u r e s used by most a u t h o r s i n c h a r a c t e r i z i n g the 4 - c o o r d i n a t e d complex. This s e a r c h has been f r u i t l e s s , as a l l such symmetric c o n f i g u r a t i o n s have c o r r e s p o n d e d to l o c a l minima w h i c h were h i g h e r i n energy than the energy quoted above f o r the " b e s t must r e g a r d the " t r u e the time average of structures in which the d i s t i n c t i o n between the i n n e r c o o r d i n a t i o n s h e l l and the o u t e r one is blurred. Narten et a l i i (39) c a r e f u l l y r e f e r t o t h e i r symmetric s t r u c t u r e as b e i n g a time average one. The 4-hydrated i o n i s , i n a s e n s e , the solution structure, and t h e r e f o r e i s o f paramount i m p o r t a n c e . The b e h a v i o r o f the energy h y p e r - s u r f a c e , d e s c r i b e d above, i s not o n l y odd i n and o f i t s e l f , but i s a t s u b s t a n t i a l v a r i a n c e w i t h the e q u i v a l e n t b e h a v i o r o f the 4-hydrated p r o t o n ( o r the 3-hydrated hydronium ion). The P o l a r i z a t i o n Model c o m p u t a t i o n s on t h i s m o i e t y found a l o c a l minimum i n the energy hyners u r f a c e which c o r r e s p o n d s to the r o u g h l y p y r a m i d a l s t r u c t u r e shown i n F i g u r e 7 a and b. ( We a l s o show, f o r completeness, i n F i g u r e 8 the extended s t r u c t u r e which c o r r e s p o n d e d to the a b s o l u t e l y l o w e s t energy w h i c h we c o u l d f i n d f o r the 4-hydrated p r o t o n . T h i s s t r u c t u r e corresponds c l o s e l y to +

(H 0) - H - (OH ) 2 2 2 2 r a t h e r than +

(H 0) - H 0 2 3 3 Both 01 and 03 a r e hydrogen bond donors i n t h i s case.) The hydronium i o n i s based on Oxygen 03 and 03 i s a " d o u b l e donor" i n f o r m i n g hydrogen bonds. I f one imagines hydrogen 6 t r a n s m u t i n g i n t o the L i t h i u m c a t i o n , one sees t h a t the l i g a n d s c o n t a i n i n g oxygens 02 and 04 a r e b o t h i n the second c o o r d i n a t i o n sphere

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o f the p s e u d o - L i t h i u m ion. Two hydrogen bonds a r e b e i n g formed. Thus, the energy h y p e r - s u r f a c e f o r the 4 - h y d r a t e d L i t h i u m c a t i o n i s not b e h a v i n g a n a l o g o u s l y t o t h e energy h y p e r - s u r f a c e f o r t h e 4-hydrated proton. Although i t i s possible to blindly continue i n v e s t i g a t i n g higher hydrates, i t would appear t o be more p r o f i t a b l e t o t u r n our a t t e n t i o n t o chemical s p e c i e s i n the L i t h i u m , Hydrogen, and Oxygen f a m i l y f o r w h i c h s t r u c t u r e s a r e a c t u a l l y known, even i f the e n e r g i e s o f t h e s e s p e c i e s are not w e l l known. Higher

Species.

H i n c h c l i f f e and Dobson (4£ ,41) r e p o r t t h e calculation o f the s t r u c t u r e o f T T - O - L i as b e i n g "almost" p l a n a r , wit k c a l / m o l . U s i n g th the minimum energy s t r u c t u r e found had an energy o f -629.3 k c a l / m o l , in suprisingly good agreement w i t h the t h e o r e t i c a l v a l u e , and a L i - O - L i a n g l e o f 153.5 degrees. The L i - 0 bond l e n g t h s were found t o be e q u a l , h a v i n g v a l u e o f 1.505 A. U n f o r t u n a t e l y , Grow and P i t z e r (42) r e p o r t t h a t L i - O - L i i s l i n e a r , w i t h a L i - 0 bond l e n g t h o f between 1.55 and 1.6 Angstrom. It is fairly c l e a r that this linear structure i s c l o s e r t o the " t r u t h " than our bent s t r u c t u r e . We o b t a i n a computed d i p o l e moment f o r t h i s m o i e t y o f 1.275 D, while Buchler et a l i i (43) r e p o r t t h a t their experimental determination pTaces an upper bound o f 0.2 D on the p o s s i b l e d i p o l e moment. Of course, there i s a b s o l u t e l y no reason that the p a r a m a t e r i z a t i o n made on an i s o l a t e d Lithium cation interacting with a w a t e r m o l e c u l e ' s oxygen atom s h o u l d be v a l i d i n the s i t u a t i o n i n w h i c h the L i t h i u m c a t i o n i s c h e m i c a l l y bound t o an o x i d e i o n . We have a l s o i n v e s t i g a t e d t h e i s o l a t e d LiOU molecule, finding an o p t i m a l energy o f -761.3 k c a l / m o l , w i t h a Li-O-H a n g l e o f 155.5 d e g r e e s , an L i - 0 d i s t a n c e o f 1.67 A and an O-H d i s t a n c e o f .968 A. The P o l a r i z a t i o n Model c o m p u t a t i o n o f the energy of the hydroxide ion yields an energy f o r t h i s s p e c i e s o f -643.1 k c a l / m o l , so t h a t the binding energy o f the LiOH m o l e c u l e i s -128.2 k c a l / m o l . An o c c u r r e n c e o f hard e x p e r i m e n t a l l y obtained e v i d e n c e f o r s t r u c t u r e s i n the aquated L i t h i u m c a t i o n sequence (44) i s c r y s t a l l o g r a p h i c evidence on the s t r u c t u r e oT

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(H

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0) 2

i n w h i c h the Lithium i o n s are surrounded by two H y d r o x i d e i o n s and two waters of h y d r a t i o n , the waters being t e t r a h e d r a l l y disposed. We have investigated the i s o l a t e d molecule of hydrated Lithium Hydroxide,, and find a minimum energy c o n f i g u r a t i o n w i t h energy -1888.8 k c a l / m o l . Figure 9 shows the u l t i m a t e d i s p o s i t i o n o f the two m o i e t i e s for this i s o l a t e d molecule. 9+ f o r 01 is 28.1 d e g r e e s . The L i - 0 2 - H a n g l e i s 106.5 degrees. The L i 01 bond d i s t a n c e i s 2.08 A, w h i l e the L i - ^ 2 bond d i s t a n c e i s 1.6 A. For the c r y s t a l , the L i - 0 ( o f water) d i s t a n c e i s 1.982 A, while the L i - 0 (of hydroxide) distance i s 1.966 A. We have a l s o i n v e s t i g a t e d the s t r u c t u r s t a r t i n g s t r u c t u r e suc c a l c u l a t i o n o f Urban e t a l i i ( 4 5 ) , where the water molecule l i e s between the hy?Troxide i o n and the Lithium cation. The resultant minimum energy structure i s p r e d i c t e d to have a h i g h e r energy, -1854.5 k c a l / m o l . A quantum m e c h a n i c a l c o m p u t a t i o n using the reversed geometry would be quite i n t e r e s t i n g now. Monte C a r l o

Simulations

The u n s a t i s f a c t o r y nature of searching for minimum energy c o n f i g u r a t i o n s f o r s t a b l e species w h i c h need not e x i s t as d i s c r e t e e n t i t i e s lends one to consider simulating larger systems. Our e x p e r i e n c e w i t h the 4-hydrated L i t h i u m c a t i o n demands t h a t one i n v e s t i g a t e the b e h a v i o r of t h i s s p e c i e s , diluted in water, under q u a s i - thermodynamic conditions. T h i s means computer s i m u l a t i o n , e i t h e r using the Monte C a r l o t e c h n i q u e or the molecular dynamics t e c h n i q u e . Logistic considerations indicate t h a t one s h o u l d start such endeavors u s i n g Monte C a r l o s i m u l a t i o n s on s m a l l systems, and we are here able to indicate a p r e l i m i n a r y study on the 4-hydrated L i t h i u m c a t i o n . The Monte C a r l o method i s a w e l l known scheme. As f o r m u l a t e d by M e t r o p o l i s e t a l i i ( 4 6 ) a c h a i n of c o n f i g u r a t i o n s o f the system i n quesTTon i s c r e a t e d a c c o r d i n g t o an a l g o r i t h m which we here d e s c r i b e as a p p l i e d t o our problem. Given a s t a r t i n g n u c l e a r configuration, one generates a new possible c o n f i g u r a t i o n by randomly moving e i t h e r a p r o t o n or an o x i d e i o n i n a random d i r e c t i o n by a random(but

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Polarization Model and the Hydrated Lithium Cation

DAVID

a

Figure 7 (a) The symmetric trihydroted hydronium ion. (b) Another view of the symmetric trihydrated hydronium ion.

Figure 8. The lowest energy structure found for the quadrihydrated proton

Figure 9. The hydrate of LiOH as found by the Polarization Model

Figure 10. A snapshot of the fourhydrated lithium cation during a 300 K Monte Carlo simulation

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Tabic 1. Pertinent Geometric Data for the Four-hydrated Proton in Two Configurations, the Symmetric One Shown in Figures 7a and 7b and the Elongated One Shown in Figure 8, and Comparable Data for the Four-hydrated Lithium Cation, Figure 6

Species

d(0 - H ) n 1

• symmetric F 0 9 4

3-6 1.224 1-6 1.357 3-5 1.077 3-7 1.076

Species

d(0 - F ) n 1

+ elongated F 0 9 4

1.278 1.288 1.079 1.004

d(0 •• Li) n

Species

Li

3-6 1-6 3-5 3-7

(F 0) 2 4

3 4 1 2

2.039 2.544 1.910 3.907

Q - 0 - 0 1 m n 1-3-2 111.28 1-3-4 124.40 2-3-4 116.47 F - 0 -F 1 m n 5 3-6 112.97

0 - 0 - 0 1 m n 1-3-2 115.91 1-3-4 122.71 F - 0 -F 1 m n 5-3-6 114.50 1-1-6 121.66 5-3-7 113.57

n - 0 - n 1

m

n

90.48 2-1- 3 2-1- 4 113.28 1-3- 4 69.02 3-4- 1 53.66 4-1- 3 57.32 Li - 0 - !' m i 99.75 1-1 1-2 109.34

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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small) amount. The energy o f t h i s new ( p o s s i b l e ) configuration i s c a l c u l a t e d and compared to the energy o f the l a s t c o n f i g u r a t i o n accepted i n t o the chain. I f the new energy i s l o w e r , one chooses as the n e x t c o n f i g u r a t i o n , t h i s new one. I f the new energy i s h i g h e r , one forms the e x p r e s s i o n - (F new

- F )/IT old

e and compares t h i s number t o a randomly chosen number in the range 0. t o 1.. (Floating point random numbers were g e n e r a t e d u s i n g SUBROUTINE G0U3, w h i l e i n t e g e r random numbers were sometimes g e n e r a t e d from floating point random numbers by m u l t i p l i c a t i o n by s u i t a b l e l a r g e number sometimes by u s i n g I n t e r n a t i o n a l Mathematical $ S t a t i s t i c a l libraries I n c . , 1977, S i x t h F l o o r , GNB B u i l d i n g , 7500 B e l l a i r e B o u l e v a r d , Houston, Texas 77036.) Should the random number be l e s s than t h e e x p r e s s i o n , t h e new c o n f i g u r a t i o n i s chosen to be the n e x t one, w h i l e s h o u l d the c o n t r a r y o c c u r then the o l d c o n f i g u r a t i o n i s chosen a g a i n . The p r o c e d u r e i s r e p e a t e d over and over again. When s t a b i l i t y o c c u r s , a v e r a g i n g over a d i s c r e t e number o f c o n f i g u r a t i o n s o f the c h a i n i s c a r r i e d out, with energies, heat c a p a c i t i e s , and d i s t r i b u t i o n f u n c t i o n s being obtained (47). It i s normal to carry out "TTonte Carlo s i m u l a t i o n s such as t h a t o u t l i n e d above on q u a s i infinite systems. T h i s "thermodynamic" l i m i t i s c a r r i e d out i n two s t e p s . First, p e r i o d i c boundary conditions a r e invoked such that should a new c o n f i g u r a t i o n f o r c e a p a r t i c l e t h r o u g h one o f t h e artificial boundary w a l l s o f t h e c o n t a i n e r , i t disappears from t h e system, and a new p a r t i c l e appears coming through the o p p o s i t e f a c e , p r o j e c t i n g i n e x a c t l y as f a r as i t s f o r m e r s e l f p r o j e c t e d o u t . We have a v o i d e d t h i s i n our s i m u l a t i o n s because the difficulty o f implementation precludes rapid assessment o f the t e c h n i q u e t o the problem at hand. For a c e n t r a l f o r c e p o t e n t i a l m o d e l , each atom i n the m o l e c u l e i s t r e a t e d i n d i v i d u a l l y , and t h e r e f o r e , some c a r e must be t a k e n t o make sure that p e r i o d i c boundary conditions do not a r t i f i c i a l l y force i o n i z a t i o n upon the w a t e r m o l e c u l e ! The second p a r t o f t h e passage t o t h e thermodynamic l i m i t i n v o l v e s assuming t h a t the system b e i n g emulated i s no more than a u n i t c e l l o f an

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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i n f i n t e a r r a y o f r e p l i c a t e s e x t e n d i n g o f f to i n f i n i t y i n the t h r e e d i m e n s i o n a l E u c l i d e a n space. It i s well known t h a t the Coulomb p o t e n t i a l f a l l s o f f too s l o w l y to a l l o w s i m p l e convergence of the r e q u i r e d sum f o r the Madelung c o n s t a n t by d i r e c t n u m e r i c a l e v a l u a t i o n . An Ewald (48) sum must be c a r r i e d o u t , and i n the case o f p o T a r i z a b l e p a r t i c l e s , a more complicated p r o c e d u r e must be f o l l o w e d (4J9, 50) . Since there a l r e a d y e x i s t s a n a s c e n t l i t e r a t u r e (51, 52) on the problem o f s i m u l a t i o n o f s m a l l d r o p T e t s , i t seems reasonable t o postpone d i s c u s s i o n of thermodynamic systems i n the context o f the P o l a r i z a t i o n Model u n t i l more e x p e r i e n c e i s gained w i t h the methods o f c o m p u t a t i o n on the embryo d r o p l e t systems to be considered here. As a p r e l i m i n a r y , we have attempted to use the Monte C a r l o method h y d r a t i o n number o i s 4. The Monte C a r l o method i s a u s e f u l a d j u n c t to energy m i n i m i z a t i o n methods f o r f i n d i n g v e r y low energy c o n f o r m a t i o n s o f m o l e c u l e s . It i s possible, by l o w e r i n g the t e m p e r a t u r e , to f o l l o w the q u a s i s t a t i c passage o f a system from a f l u i d s t a t e t o the s t a t e o f l o w e s t energy ( " s o l i d - l i k e " ) . We have run Monte C a r l o s i m u l a t i o n s on the 4-hydrated Lithium c a t i o n , l o w e r i n g the t e m p e r a t u r e whenever the system has appeared t o come to equilibrium, u n t i l we have approached the lowest temperatures reasonable f o r study ca. 50 degrees Kelvin. This i s a r a t h e r unorthodox p r o c e d u r e i n Monte C a r l o work. However, i t i s a moderately s e n s i t i v e a l t e r n a t i v e scheme to s t r a i g h t f o r w a r d energy m i n i m i z a t i o n techniques, and a c t s as a good c r o s s check t o guard a g a i n s t a l g o r i t h m i c c o n s t r a i n t s y i e l d i n g improper minimum i n the energy m i n i m i z a t i o n technique. A l l minima, including, hopefully, the g l o b a l minimum, w h i c h are c o n n e c t e d to the s t a r t i n g c o n f i g u r a t i o n by means o f o n l y one p a r t i c l e moves, are theoretically a c c e s s i b l e t o the M o n t e Carlo minimization technique, c o n t r a r y to the case o f the s t r a i g h t forward a l g o r t h m i c a l l y determined searching techniques available. Even under Monte C a r l o c o n d i t i o n s , however, no f i r m a s s u r a n c e can be made t h a t the g l o b a l minimum has indeed been d i s c o v e r e d . With lowering temperatures, i t i s r e q u i r e d that the Monte C a r l o s t e p s i z e be c o n s t a n t l y r e a d j u s t e d to a p p r o x i m a t e a 50% acceptance r a t i o (when a trial c o n f i g u r a t i o n has an energy g r e a t e r t h a n the energy o f the last configuration, a c h o i c e m u s t be made whether o r not to a c c e p t t h i s new, higher energy, 9

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

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c o n f i g u r a t i o n . The a c c e p t a n c e r a t i o i s d e f i n e d as the r a t i o o f the number o f t i m e s the t r i a l c o n f i g u r a t i o n is chosen compared to the t o t a l number o f c o n f i g u r a t i o n s a l r e a d y i n c l u d e d i n the Markov c h a i n ) . In s i m u l a t i n g s m a l l c l u s t e r s , one immmediately f a c e s a d e f i n t i o n problem c o n c e r n i n g the n a t u r e o f a cluster. Abraham (53) considers c l u s t e r s as b e i n g bounded by a p h y s i c a l " c o n t a i n i n g sphere (54) w h i l e B i n d e r (55) r e q u i r e s i n s t e a d a minimal connectedness to d e f i n e a c l u s t e r ( n o t i c e the r e b u t t a l argument o f Abraham (56) ) . We have f o u n d , as have K a e l b e r e r and E t t e r s (577 t h a t at low enough t e m p e r a t u r e s , t h e d i s t i n c t i o n i s moot, as s m a l l c l u s t e r s remain i n t a c t under e i t h e r d e f i n i t i o n . T h e r e f o r e , we have c a r r i e d out our simulations under the c o n s t r a i n t that a lithium c a t i o n i s l o c a l i z e d a t the o r i g i n . No boundary s u r f a c e i employed th lo cases, examination o shows t h a t the c l u s t e r s o f h y d r a t e d lithium cations remain i n t a c t . However, i t i s not a t a l l c l e a r t h a t t h i s same s i t u a t i o n e x i s t s at h i g h e r t e m p e r a t u r e s . In f a c t , the l o c a l i z a t i o n of the c l u s t e r s provides a d i f f e r e n t problem, one d i s c u s s e d by Wood (58) i n h i s famous r e v i e w a r t i c l e . The system s t a y s i n a p o c k e t o f c o n f i g u r a t i o n space, w i t h o u t jumping to one o f the N! e q u i v a l e n t p o c k e t s which d e s c r i b e interchanges o f water m o l e c u l e s . ( A t t h e low t e m p e r a t u r e s e n c o u n t e r e d h e r e , no i o n i z a t i o n t a k e s p l a c e , and t h e r e f o r e no i n t e r c h a n g e s o f p r o t o n s among waters occurs.) I f we have been u n f o r t u n a t e enough to choose a " p o o r " s t a r t i n g c o n f i g u r a t i o n , one i n w h i c h the ion c l u s t e r i s d i s j o i n t i n the sense o f Binder, then, we have observed that, a t high temperatures, d i s s o c i a t i o n o f the c l u s t e r o c c u r s , w h i l e a t low t e m p e r a t u r e s , we o b s e r v e e q u i l i b r a t e d s o l i d - l i k e s t r u c t u r e w h i c h are quite elongated. On the contrary, i f we s t a r t w i t h compact beginning clusters, then the simulation creates a chain o f c o n f i g u r a t i o n s which, i n l a r g e measure, keeps the c o n f i g u r a t i o n s compact. I t i s t h e r e f o r e d i f f i c u l t t o know i f one i s i n a p o c k e t o f s i n g l e c l u s t e r c o n f i g u r a t i o n s or m u l t i p l e d i s j o i n t c l u s t e r s . We have o b t a i n e d drawings o f the f i n a l c l u s t e r s , and have attempted to r e p o r t here o n l y compact c l u s t e r s , whose bond d i s t r i b u t i o n f u n c t i o n s i n d i c a t e t h a t they are b e i n g h e l d t o g e t h e r as a u n i t . Results o f Simulations We have attempted

t o ask

the q u e s t i o n ,

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

i s the

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.08

Figure 11. The bond distribution functions for the lithium-oxygen and the lithium-proton distances, taken from a 300 K Monte Carlo simulation. This function measures the fraction of named bond lengths with length between r and r + dr where dr in our case is 0.1 A. The Li-O function is shown in solid while the Li-H function is shown in dashed line.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

Polarization Model and the Hydrated Lithium Cation

DAVID

O

1.5

57

Figure 13. A snapshot of the four-hydrated complex at 50 K

2.5

3.5

4.5

Figure 14. The bond distribution functions for the lithium-oxygen and the lithium-proton distances, taken from a 50 K Monte Carlo simulation

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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coordination number of the Lithium cation in the 4-hydrated gaseous i o n 4? In o r d e r to answer t h i s q u e s t i o n , we have c a r r i e d out a s e r i e s o f s i m u l a t i o n s at descending temperatures, hoping to f o l l o w the ouasi-static passage to the minimum energy c o n f i g u r a t i o n of the complex. In F i g u r e (10) is shown a t y p i c a l c o n f i g u r a t i o n ( i n essence a s n a p s h o t ) o f the complex d u r i n g a 300 degree K e l v i n r u n . In F i g u r e (11) i s shown the bond d i s t r i b u t i o n f u n c t i o n for botTT~the L i - 0 bond l e n g t h s and the L i - I bond lengths. The bond d i s t r i b u t i o n function i s defined as the f r a c t i o n o f a l l bonds o f the type i n d i c a t e d with lengths which f a l l between r and r + dr. A snapshot o f an i n t e r m e d i a t e s i m u l a t i o n , F i g u r e ( 1 2 ) , t a k e n at 150 degrees K e l v i n shows t h a t compactness i s being achieved. In F i g u r e (13) and (14) are shown the snapshot and bon f o r a t y p i c a l 50 degre the oxygen atoms are not d i s p o s e d t e t r a h e d r a l 1 y about the L i t h i u m c a t i o n , and the c o o r d i n a t i o n number o f the c a t i o n i s not 4. R a t h e r , one water l i g a n d i s c o n s t a n t l y b e i n g f o r c e d i n t o the second c o o r d i n a t i o n s p h e r e , j u s t as i n the s t a t i c energy m i n i m i z a t i o n s . We are forced to c o n c l u d e t h a t no saddle point problems have o c c u r r e d in the static energy m i n i m i z a t i o n s which were c a r r i e d o u t ( a b o v e ) on t h i s species, and that indeed, the symmetric 4-coordination expected i n t h i s case i s a phase phenomenon, r e s u l t i n g from p a c k i n g o f the h y d r a t e d i o n i n the dense s o l v e n t w a t e r . A r e p o r t on more t r a d i t i o n a l aspects of the Monte C a r l o s i m u l a t i o n s on d r o p l e t embryos, w i t h and without a Lithium c a t i o n present, i s being prepared, and w i l l be r e p o r t e d i n the n e a r f u t u r e . T

Acknowledgements I t i s a p l e a s u r e to acknowledge the help given me by the s t a f f of the U n i v e r s i t y o f Connecticut R e s e a r c h Computer C e n t e r , with s p e c i a l thanks to P e t e r R u t i s h a u s e r and Tim F a r t i g a n . Generous g r a n t s o f computer time by the C e n t e r are gratefully acknowledged. A l l computations reported here were performed on the UCC IBM 360/65 § 370/155 computers. M o l e c u l a r drawings f o r t h i s work were p r e p a r e d using ORTEP, A Fortran Thermal-Ellipsoid Plot program f o r C r y s t a l S t r u c t u r e I l l u s t r a t i o n s , a u t h o r e d by C a r r o l l K. J o h n s o n , Oak Ridge N a t i o n a l L a b o r a t o r y , ORNL-7 (second r e v i s i o n ) , U C . 4 - C h e m i s t r y , 1 9 7 0 ORTFPII, March 15, 1971 v e r s i o n u s i n g the hidden l i n e

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

DAVID

Polarization Model and the Hydrated Lithium Cation

59

algorithm. A c t u a l p l o t t i n g was c a r r i e d o u t o f f l i n e u s i n g a n IBM 1620 w i t h a t t a c h e d p l o t t e r . Dr. Frank Stillinger's continuing a i d , encouragement, and h e l p , i s g r a t e f u l l y a c k n o w l e d g e d . Literature

Cited

(1) Born M., and Heisenberg W., Z. Physic(1924)23,388. (2) Rittner, E. S., J. Chem. P h y s . ( 1 9 5 1 ) 1 9 , 1 0 3 0 . (3) Etters R. D., Danilowicz R., and Dugan J., J. Chem. P h y s . ( 1 9 7 7 ) 6 7 , 1 5 7 0 . (4) Guido M., and Gigli G., J. Chem. Phys. (1977)66,3920. (5) Ben-Naim A., a n d Stillinger F. H., " A s p e c t s of the Statistical Mechanical Theory o f Water, in Structure and Transport Aqueous Solutions", Interscience N.Y., 1 9 7 2 . (6) Rahman A., a n d Stillinger F. H., J. Chem. Phys. (1971) 55,13336. (7) Stillinger F. H., a n d Rahman A., J . Chem. Phys. (1974)60,1545. (8) S h i p m a n L. L . , a n d Scheraga H. A., J. Chem. Phys.(1974)78,909. (9) Cambell E. S., a n d M e z e i M., J. Chem. Phys. (1977)67,2338. ( 1 0 ) P o p k i e H., K i s t e n m a c h e r H., a n d Clementi E . , J. Chem. P h y s . ( 1 9 7 3 ) 5 9 , 1 3 2 5 . (11) Campbell, E. S., a n d Mezei, M., J. Chem. Phys.(1977)67,2338. (12) Berendsen H. J . C. a n d van der Velde G. A., "Molecular Dynamics and Monte Carlo Simulation on W a t e r " , R e p o r t o n the C. E. C. A. M. W o r k s h o p , held in Orsay, 1 9 7 2 , p g . 6 3 . ( 1 3 ) L e m b e r g H. L . , a n d Stillinger F. H., J. Chem. P h y s . (.1975)62,1677. ( 1 4 ) Rahman A., Stillinger F. H., a n d L e m b e r g H., J. Chem. P h y s . ( 1 9 7 5 ) 6 3 , 5 2 2 3 . (15) Stillinger F. H., a n d D a v i d C. W., J. Chem. Phys. manuscript in preparation. ( 1 6 ) K i s t e n m a c h e r F., Popkie H., a n d Clementi F . , J. Chem. P h y s . ( 1 9 7 3 ) 5 9 , 5 8 4 2 . (17) Stillinger F. H., private communication. (18) Kittel C., "Introduction to Solid State Physic", J o h n Wiley a n d S o n s , Inc., New York, 1953, pg 165. (19) Dzidic I . , and Kebarle P., J . Phys. Chem. ( 1 9 7 0 ) 7 4 , 1 4 6 6 . ( 2 0 ) F r i e d m a n H. L . , a n d Lewis L . , J. Sol'n

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

60

COMPUTER MODELING OF

MATTER

Chem.(1976)5,445. (21) V a s l o w F., J. Phys. Chem.(1963)67,2773. (22) Eliezer I., and Krindel B., J. Chem. Phys. (1972)57,1884. (23) Kistenmacher H., Popkie H., and E. C l e m e n t i F.,loc.cit.. (24) Clementi E., and Popkie H., J. Chem. Phys.(1972)57,1077. (25) Woodin R. L., Houle F. A., and W. A. Goddard III W. A., Chem. Phys.(1976)14,461. (26) D i e r c k s e n G. H. F., and Kaemer W. P., Theo. Chim. A c t a ( B e r l ) ( 1 9 7 5 ) 3 6 , 2 4 9 . (27) S c h u s t e r P., J a k u b e t z W., and M a r i u s W., " T o p i c s in C u r r e n t C h e m i s t r y " (1975)60,1. (28) Kollman P. A. and Kuntz T. D., J. Am. Chem. Soc., (1972)94,9236. (29) Kollman P. A. an (1972)96,4766. (30) Kollman P. A., J. Am. Chem. Soc. (1977)99,,4875. (31) Kollman P. A., Accounts of C h e m i c a l R e s e a r c h , (1977)10,365. (32) Spears K. G., J. Chem. Phys. (1972)57,1850. (33) S p e a r s , K. and Kim, S. H. J. Phys. Chem.(1976)80,673. (34) Newton M. D., and Ehrenson S., J . Am. Chem. Soc. (1971) 92,497. (35) Kollman P.A., and Bender C. F., Chem. Phys. Letters (1973) 21, 271. (36) L i s c h k a , H., and Dyczmons, V., Chem. Phys. Letters,(1973)23,167. (37) Geiger A., and Hertz H. G.,J. Sol'n. Chem.(1976)5,365 (38) N a r t e n A. H., Vaslow F., and Levy H. A . , J . Chem. Phys. (1973)58,5017. (39) A. H. N a r t e n , F. Vaslow, and H. A. Levy, loc. cit. (40) Hinchliffe A., and Dobson J. C., Chem. Phys.(1975)8,166. (41) Hinchliffe A., and Dobson J . C., Mol. Phys. (1974)28,543. (42) Grow D. T., and Pitzer R. M.,J. Chem. Phys. (1977)67,4019. (43) B u c h l e r A., Stauffler J . , Klemperer W., and Wharton L., J. Chem. Phys.(1963)39,2299. (44) Gennick I., and Harmon K. M., Inorg. Chem.(1975)14,2214, and r e f e r e n c e s c o n t a i n e d t h e r e i n . (45) Urban M., Pavliks S., Kello V., and M a r d i a k o v J., Collection C z e c h o s l o v . Chem. Commun.(1975)40,587. (46) N. Metropolis, A. W. Rosenbluth, M. N. R o s e n b l u t h , A. H. Teller, and E. Teller, J. Chem.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

4.

DAVID

Polarization Model and the Hydrated Lithium Cation

61

Phys. (1953)21,1087. (47) An excellent discussion o f the Monte-Carlo method can he found in A. Ben-Naim, "Water and Aqueous Solutions", Plenum P r e s s , N.Y. 1974, pg. 73. (48) Ewald P. P.,Ann. P h y s i k ( 1 9 2 1 ) 6 4 , 2 5 3 . (49) K o r n f e l d H.,Z. P h y s i k (1924)22,27. (50) V e s e l y F. J., J. "Comp. Phys. (1977) 24,361. (51) Lee, J. K., B a r k e r , J. A., and Abraham, F. F., J. Chem.Phys. (1973)58,3166. (52) Priant, C. L., and B u r t o n , J. J., J. Chem. Phys. (1975)63,3327. (53) Abraham, F. A., J. Chem. Phys.(1974)61,1221. (54) Lee, J. K., B a r k e r , J. A., and Abraham, F. A . , J . Chem. Phys. (1973)58,3166. (55) B i n d e r , K., J. Chem. Phys., (1975)63,2265. (56) Abraham, F. A., and B a r k e r , J . A., J . Chem. Phys. (1975)63,2266 (57) K a e l b e r e r , J. Phys.(1977)66,3233. (58) Wood, W. W., " P h y s i c s of Simple Fluids", Eds. F. N. V. Temperley, J. S. R o w l i n s o n , and G. S. Rushbrooke, W i l e y Interscience, John W i l e y and Sons, I n c . , New Y o r k , 1968. RECEIVED

August 15,

1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5 Molecular Dynamics Simulation of Methane Using a Singularity-Free Algorithm S. MURAD and K. E. GUBBINS School of Chemical Engineering, Cornell University, Ithaca, NY 14853

Rather extensive compute ported f o r l i q u i d s o f linear Much l e s s work has been reported f o r l i q u i d s of n o n l i n e a r mole c u l e s , water (2) being the only exception, although some simula­ t i o n work f o r ammonia (3), benzene (4,5) and n-butane (6) has been published r e c e n t l y . Simulations o f rigid polyatomics are complicated by the a d d i t i o n a l o r i e n t a t i o n v a r i a b l e s , and a l s o because the equations that must be solved f o r the r o t a t i o n a l motion possess a s i n g u l a r i t y at θ = 0 if the usual E u l e r angles are used to represent the o r i e n t a t i o n s . This l a t t e r difficulty was overcome by Evans and Murad (7) by using quaternions i n place of E u l e r angles. T h i s r e s u l t s i n a more efficient and accurate algorithm, and y i e l d s an order of magnitude improvement i n energy conservation. A f u r t h e r i n c r e a s e i n computing speed by a f a c t o r of three to eight may be obtained by using the m u l t i p l e time step method developed by S t r e e t t et al. (8). The a p p l i c a t i o n of t h i s method to r i g i d polyatomic molecules i s described i n another paper i n t h i s volume (9). In t h i s paper we use the quaternion method o f Evans and Murad to study methane, and report p r e l i m i n a r y r e s u l t s using a truncated and s h i f t e d s i t e - s i t e Lennard-Jones p o t e n t i a l . The m u l t i p l e time step method was not used i n t h i s work, p r i n c i p a l l y because o f storage l i m i t a t i o n s on the PDP 11/70 minicomputer used. The p o t e n t i a l model s t u d i e d i n v o l v e s f i v e Lennard-Jones s i t e s f o r each molecule, l o c a t e d on the C and H n u c l e i (C-H bond length 1.10 A). Thus the p a i r p o t e n t i a l i s the sum of 25 i n t e r ­ a c t i o n s , 1 between C...C, 8 between C...H, and 16 between H...H sites. The p o t e n t i a l i s thus uCraUiU,) = I [ u ( r a 6

= 0

a ( J

) - u

o ( J

(r )] 0

r < T

Q

(1)

r > r

o where ri2 i s the v e c t o r from the center of molecule 1 to that o f

0-8412-0463-2/78/47-086-062$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5.

MURAD

AND

63

Molecular Dynamics Simulation of Methane

GUBBINS

molecule 2, = 4>i0^^ i s the o r i e n t a t i o n o f molecule i , r g i s the distance between s i t e a of molecule 1 and s i t e 8 o f molecule 2, r = 10.025 A i s the c u t o f f d i s t a n c e , and a

Q

-

W

(2)

4 e

where e g and a g are the Lennard-Jones parameters (see Figure 1). Using t h i s p o t e n t i a l model, f i v e sets o f parameters (e 3>o* $) have been used to evaluate thermodynamic p r o p e r t i e s ( c o n f i g u r a t i o n a l energy, pressure and s p e c i f i c h e a t ) . The best o f these parameter sets i s used i n making a more d e t a i l e d study o f methane. This best set gives r e s u l t s s u p e r i o r to those o f a recent study (10) u t i l i z i n g the Williams (11) s i t e - s i t e exp-6 model, and i s also s u p e r i o r to the r e s u l t s obtained by Hanley and Watts (12) using an i s o t r o p i c m-6-8 a

a

a

a

Method The quaternion method has been described i n d e t a i l i n recent p u b l i c a t i o n s CO so that we only give a b r i e f summary o f the method here. The quaternion parameters x> n, £ C can be defined i n terms of G o l d s t e i n E u l e r angles (13) by ?

X

= cos

(6/2)

n

= s i n (0/2)

c o s ( * + )/2, ^ c o s ( * - <j>)/2,

(3)

> 5

= s i n (0/2)

s i n ( * - <(>)/2,

C

= cos

sin(i|i + (J>)/2.

(0/2)

j

From (3) i t can be seen that X

2

+ n

2

+

C + C =1

(4)

2

The r o t a t i o n matrix A (13) defined by V = A • V -principal - -lab

(5)

i s given by

-S A =

2

I -2«n L

+

n

2

- C + x,

+ cx),

2(nc - 5x).

2Ux - 5n),

2

2

S - n 2

2

- c

-2(5c + nx).

2

+ x, 2

-S

2

2(nc + £x)

(6)

2(nx - SO

n

2

+ c

2

+ x

2

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

64

COMPUTER

MODELING

OF

MATTER

The p r i n c i p a l angular v e l o c i t y u>p i s r e l a t e d to the quaternions by the matrix equation W

' Px\ (7)

0 / The i n v e r s e of (7) i s simple to f i n d s i n c e the 4 x 4 matrix i s orthogonal. The equations of motion f o r these parameters are thus f r e e of s i n g u l a r i t i e s . A system o f 108 model methane molecules was s t u d i e d using standard p e r i o d i c boundary c o n d i t i o n s ( 1 ) . The i n t e r m o l e c u l a r forces and torques wer system. The torques wer Tp, using the r o t a t i o n matrix (equation ( 6 ) ) . P r i n c i p a l torques and f o r c e s F were used to evaluate the time d e r i v a t i v e s of the p r i n c i p a l angular v e l o c i t i e s and c a r t e s i a n p o s i t i o n s r of the molecules using the equations du) -77^ = T d t

/I Pa P

(a = x,y,z)

(8)

a

2

d r dP"

(9)

F/m

A t h i r d order p r e d i c t o r - c o r r e c t o r method (14) was used to i n t e ­ grate the center of mass motion (equation ( 9 ) ) , while a second order method was used f o r o r i e n t a t i o n a l equations of motion of the molecules (equations (8) and ( 7 ) ) . A l l r e s u l t s were based on 2,000 time steps of 1.47 x 1 0 " seconds, a f t e r e q u i l i b r a t i o n . The energy and pressure were c o r r e c t e d f o r the long range p o t e n t i a l c o n t r i b u t i o n i n the usual way ( 1 ) , by p u t t i n g g g = 1 f o r r g > r . The s h i f t i n the p o t e n t i a l a l s o a f f e c t s the energy, but not the pressure or s p e c i f i c heat. The energy was c o r r e c t e d f o r t h i s p o t e n t i a l s h i f t by w r i t i n g U = + U h i f t > where Ujfl) i s the value corresponding to equations (1) and (2) and obtained i n the s i m u l a t i o n , and U h i f t i s the c o r r e c t i o n , given by 15

a

a

Q

s

s

where g ( r ) i s the s i t e - s i t e c o r r e l a t i o n f u n c t i o n . n

n

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5.

MURAD

65

Molecular Dynamics Simulation of Methane

A N D GUBBINS

Energy conservation was w i t h i n about 0.75% per 10,000 time steps. One time step took approximately 25 seconds f o r a 108 molecule system on the PDP 11/70 minicomputer. Results The p o t e n t i a l o f equations (1) and (2) i n v o l v e s the para­ meter set ( e , a , ecc/ HH> Cc/°HH> ^CH* ^CH)» where ncH CcH g i the departure from the usual combining r u l e s f o r and CCH, e

c c

a

a

n

d

c c

v e

Q

CH

=

n

(a

* CH CC ke

CH

^CH CC

+

a

HH

(10)

}

(11)

HH

In t h i s work we have s t u d i e e

e

a

n

d a

a

T

i

CC^ HH CC^ HH* * and are put equal to those given by Williams (11) i n h i s para­ meter set VII. Table I shows the Lennard-Jones parameter sets used i n t h i s work. In the f i r s t f i v e sets ZQQ and OQQ are kept f i x e d , and the r a t i o s ^QQ/O^H and ecc/ HH are v a r i e d from one set to the next. The parameters f o r set 1 were obtained by p u t t i n g the Lennard-Jones o* g and e g values equal to those used by Williams (11) i n h i s parameter set VII f o r the exp-6 model. Parameter sets 2-5 show increases i n £HH o f e i t h e r 10 o r 20%, and decreases i n of e i t h e r 0, 1, o r 2%. In a l l f i v e sets ncH n d were kept f i x e d at 0.972 and 1.132, r e s p e c t i v e l y , as given by Williams parameter set VII. For each o f the 5 parameter sets given i n Table I three s t a t e points were s t u d i e d and the mean squared d e v i a t i o n s between molecular dynamics and experimental r e s u l t s (15) were c a l c u l a t e d for the c o n f i g u r a t i o n a l energy, pressure and s p e c i f i c heat. The three s t a t e points used were: (a) T = 200 K, p = 10.00 mol fc" , (b) T = 225 K, p = 18.50 mol Z" , (c) T = 125 K, p = 25.27 mol ST . The f i r s t o f these c o n d i t i o n s i s i n the moderately dense super­ c r i t i c a l r e g i o n , the second i s i n the dense s u p e r c r i t i c a l r e g i o n , and the t h i r d i s i n the dense l i q u i d r e g i o n . Table I I gives the root mean squared d e v i a t i o n s found f o r each parameter s e t . Set 5 gave the best r e s u l t s . For t h i s parameter set s i x a d d i ­ t i o n a l s t a t e c o n d i t i o n s were s t u d i e d . The r e s u l t i n g molecular dynamics r e s u l t s are compared with experiment i n Table I I I . The molecular dynamics r e s u l t s shown i n Table I I I may be used to r e s c a l e the values of SQQ and OQQ> keeping f i x e d CC/ HH> CC/ HH> ^CH ^CH* ^ t h o d o f c a r r y i n g out t h i s r e s c a l i n g i s i l l u s t r a t e d ( f o r two s t a t e c o n d i t i o n s ) i n Figure 2. For a s i n g l e s t a t e c o n d i t i o n (e.g. 1 i n Figure 2) i t i s p o s s i b l e to f i n d a set o f (OQQ CQQ) values that give exact agreement between molecular dynamics and experiment f o r the c o n f i g u r a t i o n a l e

a

a

a

1

1

e

G

a

a

a

n

d

T

h

1

e m

9

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

66

COMPUTER

MODELING

OF MATTER

Figure 1. Site-site and center-center distances for two CH, molecules at orientations o> and w t

t

2

Table I P o t e n t i a l Parameter Sets Studied

1 2 3 4 5 Rescaled 5

cc

CH (K)

HH (K)

"CH (X)

HH (A)

350 350 350 350 350

023 023 008 008 2.995

2.870 2.870 2.841 2.841 2.813

48.760 48.760 48.760 48.760 48.760

20.690 21.700 21.700 22.665 22.665

850 535 535 220 220

3.350

2.995

2.813

51.198

23.798

8.631

CC (A)

Parameter Set

e

/k

(K)

£

/ K

E

/ K

The b o n d l e n g t h i n parameter sets 1-5 i s kept f i x e d at 1.10 A. The parameter s e t " r e s c a l e d 5" a l s o has t h i s same value o f the bond l e n g t h , s i n c e the r e s c a l e d a's are unchanged. o

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5.

MURAD

AND

Molecular Dynamics Simulation of Methane

GUBBINS

Table I I Root Mean Squared Deviations For Models Studied

Root Mean Squared Deviation"*"

Parameter Set

C

U /NkT

P/pkT

Cy/Nk

1 2 3 4 5

0.45 0.41 0.26 0.18 0.22

0.46 0.41 0.34 0.21 0.14

0.43 0.35 0.44 0.37 0.37

0.45 0.40 0.30 0.21 0.21

Rescaled 5

0.08

0.07

0.22

0.10

V

= C V

- C

V

i d g

a

s

, where C

V

and C

i d g

T o t a l ft

a

s

are a t the

v

V

same T. We expect the e r r o r s i n our MD values to be c

no greater than 0.03 f o r U /NkT and 0.08 f o r P/pkT.

ft

For the t o t a l d e v i a t i o n , energy, pressure and s p e c i f i c heat were given the weights 4:2:1.

Table I I I Results f o r Parameter Set 5 Before R e s c a l i n g

M „ Temperature Density (K) (mol I ) n

C

U /NkT MD

Ex£

r

P/pkT MD

Ex£

C /Nk v MD Ex£

198 228 250

10.00 10.00 10.00

-1.49 -1.70 0.48 0.35 0.37 0.70 -1.28 -1.45 0.70 0.55 0.15 0.55 -1.13 -1.25 0.77 0.66 0.12 0.39

223 248 276

18.50 18.50 18.50

-2.45 -2.59 0.92 0.80 0.32 0.39 -2.15 -2.28 1.22 1.08 0.37 0.37 -1.92 -1.99 1.45 1.25 0.52 0.40

123 151 182

25.27 25.27 25.27

-6.68 -6.84 -0.10 0.0 0.55 0.85 -5.28 -5.41 1.23 1.15 0.73 0.98 -4.27 -4.33 2.19 1.91 0.72 0.79

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

67

68

COMPUTER

MODELING O F MATTER

Figure 2. Rescaling the potential parameters acc and e r, keeping the ratios VCC'-VCH'-VHH and tcc'>*cH-*nHfixed.Curves 1 and 2 correspond to different state conditions. C

Table IV Results f o r Parameter Set 5 A f t e r R e s c a l i n g

c

U /NkT

r

P/pkT

C /Nk v MD Exp

Temperature (K)

Density (mol J T )

MD

Exp

MD

Exp

208 239 263

10.00 10.00 10.00

-1.49 -1.28 -1.13

-1.59 -1.34 -1.12

0.48 0.70 0.77

0.41 0.60 0.71

0.37 0.15 0.12

0.70 0.44 0.35

234 260 290

18.50 18.50 18.50

-2.45 -2.15 -1.92

-2.45 -2.15 -1.92

0.92 1.22 1.45

0.90 1.13 1.34

0.32 0.37 0.52

0.37 0.37 0.43

129 158 191

25.27 25.27 25.27

-6.68 -5.28 -4.27

-6.52 -5.15 -4.18

-0.10 1.23 2.19

0.30 1.35 2.20

0.55 0.73 0.72

0.89 0.96 0.80

1

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

5.

MURAD

Molecular Dynamics Simulation of Methane

A N D GUBBINS

O-O 100

180

260

340

T(K)

Figure 3. Configurational internal en&J from molecular dynamics (points) and experiment (lines)

eT

1

1 00

i

180

1 T(K)

i

260

1

1—

3 40

Figure 4. Pressure from molecular dynamics (points) and experiment (lines)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

69

70

COMPUTER

MODELING

OF MATTER

energy. This y i e l d s a l i n e on a p l o t of a^ vs. £QC* A second, but d i f f e r e n t , s e t o f (OQQ, cc) values i s obtained by equating molecular dynamics and experimental values f o r the pressure at the same s t a t e c o n d i i t o n . T h i s y i e l d s a second l i n e on the CC * CC P l « The i n t e r s e c t i o n o f these two l i n e s gives a unique s e t o f ( QQ, values f o r the s t a t e c o n d i t i o n s t u d i e d . By t r e a t i n g the molecular dynamics data f o r the other s t a t e c o n d i t i o n s i n a s i m i l a r way i t i s p o s s i b l e to estimate a best set o f (GQQ, cc) l from the r e s u l t i n g " i n t e r v a l o f c o n f u s i o n " on a OQQ V S . €QQ p l o t . T h i s best s e t of r e s c a l e d parameters i s given i n Table I , and the r e s u l t i n g mean squared d e v i a t i o n s are given i n Table I I . Table IV and Figures 3 and 4 compare the molecular dynamics values with experiment f o r t h i s rescaled p o t e n t i a l . The molecular dynamics r e s u l t s found from parameter s e t 1 are almost the same as those found f o r the Williams p o t e n t i a l using h i s s e t V I I , and f o r the r e s c a l e d Lennard-Jone s t a n t i a l l y b e t t e r , as seen from Table I I . We p l a n to study f u r t h e r refinements of t h i s p o t e n t i a l model, i n c l u d i n g the a d d i t i o n o f an octupole moment, and we are a l s o i n the process of i n v e s t i g a t i n g other p r o p e r t i e s , i n c l u d i n g the o r i e n t a t i o n a l c o r r e l a t i o n f u n c t i o n s , s e l f - d i f f u s i o n c o e f f i c i e n t s , and time correlation functions. c

e

a

v s

e

o t

0

e

v

a

u

e

s

Acknowledgments We thank the American Gas A s s o c i a t i o n , the N a t i o n a l Science Foundation, and the Petroleum Research Fund, as administered by the American Chemical S o c i e t y , f o r support o f t h i s work. L i t e r a t u r e Cited 1.

S t r e e t t , W. B. and Gubbins, K. E., Ann. Rev. Phys. Chem., (1977), 28, 373.

2.

S t i l l i n g e r , F. H., Adv. Chem. Phys., (1975), 12, 107.

3.

McDonald, I . R. and 64, 4790.

4.

Kushick, J . and Berne, B. J . , J . Chem. Phys., (1976), 64, 1362.

5.

Evans, D. J . and Watts, R. O., Mol. Phys., (1976), 32, 93.

6.

Ryckaert, J . P. and Bellemans, 30, 123.

Klein,

M. E., J . Chem. Phys.,

(1976),

A., Chem. Phys. L e t t . ,

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(1975),

5.

MURAD

A N D GUBBINS

Molecular Dynamics Simulation of Methane

71

7.

Evans, D. J., Mol. Phys., (1977), 34, 317; Evans, D.J.and Murad, S., ibid., (1977), 34, 327.

8.

S t r e e t t , W. B., T i l d e s l e y , D. J . and (1978), 35, 639.

Saville,

G., Mol. Phys.,

9.

S t r e e t t , W. B., T i l d e s l e y , D. J. and t h i s volume.

Saville,

G., (1978),

10.

Murad, S., Evans, D. J., Gubbins, K. E., S t r e e t t , W. B. and T i l d e s l e y , D. J., Mol. Phys., (1978), submitted.

11.

W i l l i a m s , D. E., J. Chem. Phys.,

12.

Hanley, H. J. M. and Watts, R. O., Mol. Phys., 1907; Aust. J . Phys., (1975), 28, 315.

13.

G o l d s t e i n , H., " C l a s s i c a Wesley, (1971).

14.

Gear, C. W., "Numerical Initial Value Problems i n Ordinary Differential Equations", P r e n t i c e - H a l l , Englewood Cliffs (1971).

15.

Goodwin, R. D., "The Thermodphysical P r o p e r t i e s o f Methane from 90 to 500 K at Pressures to 700 Bars", N.B.S. T e c h n i c a l Note No. 653 (National Bureau o f Standards, Washington, DC, 1974).

RECEIVED

(1967), 47, 4680. (1975), 29.,

August 15, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

6 Structure of a Liquid-Vapor Interface M. RAO and B. J. BERNE Columbia University, New York, NY 10027

A l i q u i d c o e x i s t s wit critical temperature Tc. separated by a small t r a n s i t i o n region c a l l e d the i n t e r f a c e . Recently there has been much t h e o r e t i c a l e f f o r t devoted t o an understanding of p r o p e r t i e s o f the interface(1,2,3,4). I t i s a l s o p o s s i b l e now to shed some l i g h t on the s t r u c t u r e and dynamics o f the i n t e r f a c e s using computer simulation(5,6,7,8). With t h i s information one can begin t o evaluate v a r i o u s t h e o r e t i ­ c a l approximations and provide a q u a n t i t a t i v e framework f o r the microscopic phenomenology o f the l i q u i d - v a p o r i n t e r f a c e . In this paper we present some r e s u l t s on the l i q u i d - v a p o r i n t e r f a c e obtained from a Monte Carlo s i m u l a t i o n . Our model system c o n s i s t s o f 2 0 4 8 p a r t i c l e s , i n t e r a c t i n g via a classical mechanical Lennard-Jones (6,12) p o t e n t i a l (Mr)

= V(r) - V ( r )

0

< r < r

=0

r

Q

Q

where V(r) = 4 e [ ( ^ )

1 2

6

- (^) ] and r

Q

Q

< r

= 2.5a.

The p a r t i c l e s are

placed i n a f u l l y p e r i o d i c box o f s i z e 1 4 . 7 a * 1 4 . 7 a * 2 5 . 1 a . The d e t a i l s o f c r e a t i n g an inhomogeneous system without any ex­ t e r n a l f i e l d i n t h i s p e r i o d i c box are presented elsewhere (5^) . The s t a r t i n g c o n f i g u r a t i o n thus generated i s a two sided f i l m with l i q u i d density n^ i n the middle o f the box and vapor d e n s i t y n^ on e i t h e r s i d e .

Using the standard Metropolis

scheme the f i l m

i s e q u i l i b r a t e d a t a temperature o f 1 1 0 K. A step s i z e o f 0 . 2 a i s used f o r the random walk which gives an acceptance r a t i o o f 0.5.

At e q u i l i b r i u m the vapor d e n s i t y has an average value o f 0 . 0 5 and the l i q u i d d e n s i t y has an average value o f 0 . 6 5 ( r e ­ duced u n i t s are used throughout). The symmetrized d e n s i t y pro­ f i l e i s shown i n f i g u r e 1 with dots. The o r i g i n i s chosen t o be

0-8412-0463-2/78/47-086-072$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

1

X

o

-6.0 -5.0

1

X

o

1 -4.0

X

O

x

°

-3.0

1

X

1

X

0

-2.0

o

-1.0

1

O X

z

C)

0.2 -

O 0.4 -

0.6 -

X

0.8 •

1.0 -

1.2 -

o

X

1.0

1

o

X

0

1 2.0

X X X

o o o 1 1 3.0 4.0

X

X

o 1 5.0

Figure 1. Density and structure factor profiles of a planar sheet of Lennard-Jonesium fluid in equilibrium with its own vapor at 110 K. The origin is chosen to be the Gibbs equimolecular dividing surface. The curves are obtained from Monte Carlo simulation using one million configurations. Circles denote the density profile and the crosses denote the structure factor, r.

-7.0

1

X

°

O 0 n (z) XX s(k,z

14 -

74

COMPUTER

the Gibbs equimolecular L z

d i v i d i n g surface given

MODELING OF

MATTER

by

(n - n )

9_

= 2

g

(n

-

£

n) g

where i s the length of the box i n Z d i r e c t i o n and n i s the average d e n s i t y (=N/V). The two components of the pressure ten­ sor ( Z ) (the l o n g i t u d i n a l component) and P (Z) (the transverse P

N

T

component) are determined using 1 m i l l i o n c o n f i g u r a t i o n s gen­ erated during the random walk. The surface t e n s i o n y i s obtained from the pressure tensor L

Z/2

Y = J

[P (Z) - P ( Z ) ] d Z N

T

" Z/2 L

The value of y obtained from the s i m u l a t i o n i s 0 . 4 2 reduced u n i t s . The s u r f a c e t e n s i o n y i n curved i n t e r f a c e s i s r e l a t e d to the y i n a plane i n t e r f a c e £y Tolman(j)) Y

= Y/d

+

26/r)

c where r i s the r a d i u s of the curved i n t e r f a c e and 6 i s the cur­ vature dependence d i s t a n c e . The d e t a i l s o f e s t i m a t i n g 6 from computer s i m u l a t i o n are presented e l s e w h e r e ( 1 0 ) . The value obtained f o r 6 i s 1 . 0 a . In f i g u r e 1 we a l s o present the transverse s t r u c t u r e f a c t o r S (k,Z) p r e v i o u s l y s t u d i e d ( 6 ) i n a s i m i l a r system a t lower temp­ e r a t u r e . S (k,Z) i s defined as T

T

N(AZ)

S (k,Z) = < T

I e x p [ - i k - ( r . - r .) ] >/ i,j=l 1

where k =

0)

and

(0,

3

p a r a l l e l to the s u r f a c e , i and

j

running over the N ( A z ) atoms i n a s l i c e of volume L x L * A z centered at Z. The enhancement of S (k,Z) i n the i n t e r f a c e r e g i o n i s a t t r i b u t e d to the c a p i l l a r y waves that are thermally a c t i v a t e d . The genesis of the s i n g u l a r low-k transverse be­ h a v i o r has been d i s c u s s e d r e c e n t l y by Wertheim(_2) , Weeks (1) and Kalos, Percus and R a o ( 6 ) . The emerging p i c t u r e of the i n t e r f a c e from these analyses i s t h a t the t r a n s i t i o n from l i q u i d to vapor i s q u i t e abrupt i n the i n t e r f a c e — of the order of one d i a ­ meter — but t h i s i n t e r f a c e f l u c t u a t e s markedly i n space and time. I t i s these f l u c t u a t i o n s that broaden the i n t r i n s i c den­ s i t y p r o f i l e which i s q u i t e sharp. T h i s broadening a l s o depends on the s i z e of the system. Such a broadening has r e c e n t l y been observed i n computer s i m u l a t i o n ( 8 ) . T

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

6.

RAO A N D BERNE

Structure of a Liquid-Vapor

Interface

75

I t i s now p o s s i b l e t o simulate not only plane i n t e r f a c e s but a l s o s p h e r i c a l d r o p l e t s i n e q u i l i b r i u m with vapor(11). The s t r u c t u r e and dynamics o f these i n t e r f a c e s should throw some l i g h t on the phenomenon o f n u c l e a t i o n .

Abstract

The structure of the interface of an argon like fluid in equilibrium with its own vapor at 110 K is studied using the Monte Carlo method. The geometry of the interface is chosen to be planar and both longitudinal and transverse correlations are investigated. The longitudinal density profile shows no significant structure. From the measurement of the pressure tensor, the surface tension γ and its curvature dependence distance δ are determined. The transverse correlations exhibit very long range order in the interfac behavior. Literature Cited 1.

Weeks, J. D., J. Chem. Phys. (1977) 67, 3106.

2.

Wertheim, M. S., J . Chem. Phys. (1976) 65, 2337.

3.

L o v e t t , R. A., DeHaven, P. W., Viecelli, P., J. Chem. Phys. (1973) 58, 1880.

4.

Singh, Y. and Abraham, F. F., J. Chem. Phys. (1977) 67, 537.

5.

Rao, M. and Levesque, D., J. Chem. Phys. (1976) (65, 3233.

6.

Kalos, M. H., Percus, J . K. and Rao, M., Jour. S t a t . Phys. (1977) 17, 111.

7.

Miyazaki, J . , Barker, J . A. and Pound, G. M., J. Chem. Phys. (1976) 64, 3364.

8.

Chapela, G. A., S a v i l l e , G., Thompson, G. M. and Rowlinson, J . J . , J. Chem. S o c i e t y , Faraday T r a n s a c t i o n s I I (1977) 73, 1133.

9.

Tolman, R. C., J. Chem. Phys. (1949) 17, 333.

J.

J.

and Buff, F.

10. Rao, M. and Berne, B. J. (to be p u b l i s h e d ) . 11. Rao, M., Berne, B. J. and Kalos, M. H., J. Chem. Phys. (1978) 68, 1325. RECEIVED

August 15, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

7 Computer Simulation of the Liquid-Vapor Surface of Molecular Fluids S. M. THOMPSON and K. E. GUBBINS School of Chemical Engineering, Cornell University, Ithaca, NY 14853

The r e s u l t s describe program designed to stud vapor surfaces of systems composed o f polyatomic s p e c i e s . Both computer s i m u l a t i o n and p e r t u r b a t i o n theory methods are the t o o l s used to o b t a i n these r e s u l t s . Here we report molecular dynamics (MD) computer simulations o f two homonuclear diatomic l i q u i d s , each at a temperature in the r e g i o n of the triple p o i n t . The l i q u i d - v a p o r s u r f a c e of systems of monatomic molecules has been the subject o f considerable i n v e s t i g a t i o n (1-7) i n the past few years, but as f a r as we are aware, t h i s is the first work of t h i s type on molecular s p e c i e s . This work is initially d i r e c t e d towards o b t a i n i n g both e q u i ­ l i b r i u m and non-equilibrium p r o p e r t i e s in the inhomogeneous s u r f a c e zone f o r one component systems. E q u i l i b r i u m p r o p e r t i e s i n c l u d e the d e n s i t y - o r i e n t a t i o n profile, which provides i n f o r ­ mation on p r e f e r r e d o r i e n t a t i o n s ( i f any) i n the surface zone, and s u r f a c e tensions and energies. Non-equilibrium p r o p e r t i e s i n c l u d e t r a n s l a t i o n a l and r e - o r i e n t a t i o n a l v e l o c i t y a u t o c o r r e l a ­ t i o n functions and t h e i r a s s o c i a t e d memory f u n c t i o n s , l e a d i n g to information on the d i f f u s i o n o f molecules both perpendicular to and p a r a l l e l to the plane o f the i n t e r f a c e . Here only e q u i l i b ­ rium p r o p e r t i e s are presented. Future extensions i n c l u d e the study o f binary mixtures of molecules of v a r y i n g complexities and the behaviour of r e l a t i v e l y massy s u r f a c t a n t molecules i n the surface region. I.

The Simulation

Each system c o n s i s t s o f 216 molecules confined to square prisms o f dimensions x , V L ( = X ) and Z ( > X L ) with the usual p e r i o d i c boundary c o n d i t i o n s i n the (xy) plane of the s u r f a c e . The l i q u i d was confined to the centre o f the c e l l with vapor phases a t e i t h e r end. The two surfaces are s t a b l e without the i n c l u s i o n o f e x t e r n a l forces (2,_3>4_»j3) . I n i t i a l l y the center of mass i s f i x e d i n the center ( z = z / 2 ) o f the c e l l . The L

l

L

C 0 M

L

0-8412-0463-2/78/47-086-076$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

7.

THOMPSON AND

Liquid-V apor Surface of Molecular Fluids

GUBBINS

77

r

t r a n s l a t i o n a l and angular v e l o c i t i e s o f the molecules a r e r a n ­ domly chosen subject to the c o n s t r a i n t s that the centre o f mass has no r e s u l t a n t t r a n s l a t i o n a l or angular v e l o c i t y and that the d e s i r e d k i n e t i c energy i s obtained, t h i s being d i v i d e d between t r a n s l a t i o n and r o t a t i o n a l according to the e q u i p a r t i t i o n law. Random o r i e n t a t i o n s were assigned. The molecular centers o f mass are assigned coordinates on an f . c . c . l a t t i c e s t r u c t u r e o f the appropriate l i q u i d d e n s i t y , while the 'vapor phases a r e 'empty . As the s i m u l a t i o n proceeds, the l a t t i c e s t r u c t u r e melts and a vapor phase develops. When the r e s u l t a n t density p r o f i l e has become s t a b l e , t h i s i n i t i a l e q u i l i b r a t i o n period ( t y p i c a l l y 10* i n t e g r a t i o n steps) i s r e j e c t e d and the run proper begins. A vapor molecule which attempts to e x i t the c e l l by means o f one of the end w a l l s i s returned to the c e l l by simply bouncing i t o f f the w a l l . The low vapor d e n s i t y ensures that t h i s procedure r e s u l t s i n t o t a l l y n e g l i g i b l e d i s t o r t i o n s to the surface s t r u c ­ ture. The c e l l s were eac ( Z L = 19a), and the widt twice the p o t e n t i a l c u t - o f f d i s t a n c e . This l a t t e r quantity was 2.5 molecular diameters plus the molecular elongation (see below). Thus the c e l l widths were 5.6584 and 6.2160 molecular diameters f o r N and C l r e s p e c t i v e l y . Each f l u i d was modelled by means o f a s i t e - s i t e or atomatom p o t e n t i a l (8). Each molecule i s assumed to c o n s i s t o f two i n t e r a c t i o n centers (commonly assumed p o s i t i o n a l l y c o i n c i d e n t with the atomic n u c l e i ) . The intermolecular p o t e n t i a l i s then the sum o f four s h i f t e d Lennard-Jones (12,6) i n t e r a c t i o n s (see f i g u r e 1) 1

1

2

2

4 u(r u)ia>2) = 12

I ^JS^K^ K l

^

=

where \jS

- \j

M

M

~ LJ u

( r

c>

+

U

i j

(

r

c

)

(

r

c

"

r

)

<> 2

and ^ ( r )

- 4e[(a/r)

x a

6

- (a/r) ]

(3)

Where r i s the p o t e n t i a l c u t - o f f d i s t a n c e , the prime denoting d i f f e r e n t i a t i o n , z i s the w e l l depth and a the molecular c o l l i s i o n diameter o f the Lennard-Jones i n t e r a c t i o n , co^ = { S ^ ^ } are the p o l a r angles s p e c i f y i n g the o r i e n t a t i o n o f molecule i , and = i - 2- The f i n a l term i n (2) ensures that the p o t e n t i a l and i t s d e r i v a t i v e a r e continuous a t the c u t - o f f d i s t a n c e , l e a d i n g to b e t t e r dynamics. The four atom-atom distances are given by: c

12

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

78

COMPUTER

r! - r f 2

r

2

= r

i

3

= ri

2

X

2

12

2

2r

2

2

= r

MATTER

2

+ £| + 2 r ( £ i C O S 6 i + £ cos 6 ) + 2*ifc f(wiw )

2

2

OF

2*f + 2£iri (cos 6 i - cos 0 ) - 2£if(u)i )

2

2

r

2

MODELING

2

i 2

2

2

2

2

(£ cos 9 i + £iCos 0 ) + 2£i£ f (o)ia) ) 2

2

2

2

2£| - 2 r i £ ( c o s 0 i - cos 0 ) - 2fc|f (u>i(i>) 2

2

2

2

(4) with f(u)iU) ) = cos 0 i cos 0 2

+ s i n 0i s i n 0

2

2

cos <()i

2

(5)

The lengths Hi and £ are given i n terms o f the atomic masses mi and m and the bond length or e l o n g a t i o n L as 2

2

*i

= (m /M)L

£

= (mi/M)L

2

2

(

where M = mi + m . The p o t e n t i a l parameters and Powles (9)

6

)

2

e/K

used f o r N

2

are those of Cheung

= 37.30 K

B

a = 0.3310

ram

(7)

L/a = 0.3292 where Kg i s Boltzmann's constant. The parameters f o r C l are from the work o f Singer et a l . 2

(10) e/K

= 173.5 K

B

a = 0.3353 mm

(8)

L/a = 0.608 The center of mass motion i s found by s o l v i n g the equations

r

T

1

±

= M"

(9)

±

where r ^ i s the center of mass v e c t o r o f molecule i , ¥± i s the force on i due to a l l the other molecules and the dots i n d i c a t e d i f f e r e n t i a t i o n with respect to time. A t h i r d order p r e d i c t o r c o r r e c t o r method was used. The r o t a t i o n a l motion i s found by s o l v i n g the equations 1

w. = I " T -ip -ip 4

where

and

(10)

are the p r i n c i p a l angular v e l o c i t y of and

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

7.

T H O M P S O N A N D GUBBINS

Liquid-Vapor

Surface of Molecular Fluids

79

torque on molecule i r e s p e c t i v e l y , and I i s the moment of i n e r t i a . The o r i e n t a t i o n s were expressed by means of the quaternion para­ meters (11), and the s o l u t i o n of (10) i n terms of these para­ meters i s discussed i n (11). The use of quaternions to represent the o r i e n t a t i o n s gives s i n g u l a r i t y - f r e e equations of r o t a t i o n a l motion with b e t t e r r e s u l t a n t energy conservation. The equations of motion were i n t e g r a t e d with a reduced time step of 0.0015 [ t * = t ( c M / a ) * ] . In r e a l u n i t s t h i s i s 4.388 x 10" and 3.526 x 1 0 " s f o r N and C l r e s p e c t i v e l y . T o t a l energy was not e x a c t l y conserved, but o s c i l l a t e d around a f i x e d value with a p e r i o d of about 3000 time steps and an amplitude of about 0.2%. This behaviour replaces the slow d r i f t i n energy found when the i n t e r m o l e c u l a r f o r c e i s d i s c o n ­ tinuous at the c u t - o f f r a d i u s , that i s , the f i n a l term i n (2) i s omitted. The c a l c u l a t i o n s were performed on a PDP 11/70 computer. The computer programs wer 28.5 K 16-bit words of mented i n which molecules were ordered according to the height coordinates of t h e i r centers of mass. This was found to i n ­ crease computing speed by ^30%. Execution speed was 12.5s per step f o r the N s i m u l a t i o n . C o n f i g u r a t i o n s and t r a n s l a t i o n a l and angular v e l o c i t i e s were stored on d i s k s f o r f u r t h e r a n a l y s i s . 2

15

1 5

2

2

2

II.

Properties Calculated

The s u r f a c e t e n s i o n Y i by means of the equation s

m

was

c a l c u l a t e d during the s i m u l a t i o n

r

9 r

:=i K

K

where i and j are molecular s u b s c r i p t s , A i s the surface area and the r are i m p l i c i t l y assumed to r e f e r to the molecular p a i r (i,j). The sum over s i s over a sequence of N time steps. The p e r t u r b a t i o n provided by s h i f t i n g the Lennard-Jones p o t e n t i a l i s allowed f o r i n the s i m u l a t i o n , thus making the c a l c u l a t i o n s apply to a truncated diatomic Lennard-Jones f l u i d . The p o t e n t i a l s h i f t r e s u l t s i n a s h i f t i n the c o e x i s t i n g l i q u i d d e n s i t y (7) which i s not allowed f o r by t h i s method. The long-range t a i l of the p a i r p o t e n t i a l may be approximately taken i n t o account by adding the c o r r e c t i o n Y t r > c a l c u l a t e d as i n Appendix B of ( 7 ) . The d e n s i t y - o r i e n t a t i o n p r o f i l e p(z,0) gives the p r o b a b i l i t y density f o r f i n d i n g molecule at height z with the molecular a x i s at an angle 9 to the z a x i s (14). I t may be w r i t t e n K

s

p(z,0) =

00 I p (z)P.(cos 6) £=0 * 36

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(12)

80

COMPUTER

MODELING OF

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

MATTER

7.

Liquid-Vapor

T H O M P S O N A N D GUBBINS

Surface of Molecular Fluids

where P$,(x) i s a Legendre Polynomial. is 00

TT p

tot

( z )

=

/o

The

p ( z

'

9 )

S

i

n

6 d e

=

J

81

t o t a l density p r o f i l e 26

p Q

*

( z )

2 F + T

=

2 p

o

( z )

(13) where 6^ i s a Kronecker d e l t a and the i n d i v i d u a l p^(z) can be c a l c u l a t e d from the equations Q

p (z) £

1 ( 2 * + 1) «

P^(cos 0 ) ±

»

coefficients

/(0.1 Aa)

(14)

where 9^ i s the angle between the molecular a x i s of molecule i and the v e r t i c a l , and the average symbolized by « » i n (14) i s over a l l molecules with t h e i r centers of mass i n a s l i c e of thickness 0.1 a (and volume 0.1 Aa) about the i n d i c a t e d z coordinate over the e n t i r molecules, symmetry r e q u i r e odd I are zero. C o e f f i c i e n t s c a l c u l a t e d during the s i m u l a t i o n s were £=0, and 4. III.

Results

Averages evaluated during the simulations are given i n Table 1. The c o e x i s t i n g l i q u i d d e n s i t i e s are low as has been observed p r e v i o u s l y (7). This i s due to the s h i f t i n the p a i r p o t e n t i a l . The surface tensions are i n good (5%) agreement with experiment, the n i t r o g e n value being high w h i l e the c h l o r i n e value i s low. The surface thicknesses are defined i n terms of the gradient of the t o t a l d e n s i t y p r o f i l e at the Gibbs d i v i d i n g surface (7). The values of surface thickness shown i n Table 1 d i f f e r s i g n i f i c a n t l y from the values f o r a monatomic f l u i d under equivalent c o n d i t i o n s (7). The t o t a l density p r o f i l e s are p l o t t e d i n F i g u r e s 2 and 3 f o r N and C l r e s p e c t i v e l y . These are the averages f o r the two surfaces i n the box i n each case; the p r o f i l e s f o r the two surfaces were found to be i d e n t i c a l w i t h i n s t a t i s t i c a l e r r o r . The p r o f i l e s f o r N and C l are s i m i l a r i n t h e i r major d e t a i l s : the ' s t r u c t u r e observable i n the l i q u i d region changed i n form during the s i m u l a t i o n and i s not considered s i g n i f i c a n t . The r e a l - t i m e duration of the current simulations i s s u b s t a n t i a l l y l e s s than that of the simulations of monatomic species published to date because of the need to consider r o t a t i o n a l motion, l e a d i n g to higher estimated e r r o r i n the l i q u i d d e n s i t i e s . In Figure 4 i s p l o t t e d the second density harmonic c o e f f i c i e n t p ( z ) f o r C l at 172 K, and i n Figure 5 the d e n s i t y - o r i e n t a t i o n p r o f i l e at the z values i n d i c a t e d by * i n Figure 4. The non­ zero value i n the bulk l i q u i d i s s t a t i s t i c a l n o i s e : the co­ e f f i c i e n t here d i f f e r e d s u b s t a n t i a l l y i n both box halves and a l s o o s c i l l a t e d as the s i m u l a t i o n progressed, while the peaks at 2

2

2

2

1

2

2

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Figure 4. Second density harmonic coefficient (z) for Cl at 172.0 K. The arrow locates the Gibbs dividing surface. Pi

2

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Liquid-Vapor

T H O M P S O N A N D GUBBINS

Surface of Molecular Fluids

0.3 \

X

\

A>*(z,0) 0.2

/ ^

^

z-z = •0.30a 0

^\z-z =-0.05a0

0.1

0.0' I 0

11

1

7 T / 4

7 T / 2

1

37T/4

1

7

T

9 Figure 5. Density-orientation profile (z) + pt(z)?t(cos 6) for Cl> at 172.0 K po

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

84

COMPUTER MODELING

Table I

Averages Obtained

During

OF

MATTER

Simulations

N2_

Cl

2

40.00 x 10

3

Number of time steps

42.75

Temperature, K

66.52 + 2.01

172.0

0.608 + 0.024

0.526 ± 0.023

Surface thickness , d/a

1.79

1.14

L o c a t i o n o f Gibbs d i v i d i n g surface, z / a

4.20

4.00

Surface t e n s i o n ,

11.98

L i q u i d d e n s i t y , p*^

s

i

m

X

10

3

± 5.0

Q

JnT

x

2

10

+ 0.44

37.35 ± 0.83

+ 3

Experimental c o e x i s t i n g l i q u i d density, P* (12) L,exp —

0.664 a t 66.5 K

0.55 at 172.0 K

Experimental s u r f a c e t e n s i o n , Jm" x 1 0 (13)

11.41 at 66.5 K

39.06 at 172.0 K

2

3

z ^ 4a and z ^ 5a i s genuine s t r u c t u r e , being free o f these features. The corresponding curve f o r N i s not p l o t t e d because of poor s t a t i s t i c s f o r t h i s n e a r - s p h e r i c a l molecule. A pro­ nounced tendency to adopt p r e f e r r e d o r i e n t a t i o n s i s i n d i c a t e d , t h i s tendency being height dependent. In the l i q u i d phase at z - z = 0.85a (corresponding to the maximum i n p ( z ) ) the mole­ cules have a tendency to o r i e n t with t h e i r axes v e r t i c a l ( 9 = 0 ) , while at the Gibbs s u r f a c e ( c l o s e to the minimum i n p ( z ) ) a r e v e r s a l takes place and the molecules bend to o r i e n t with t h e i r axes p a r a l l e l to the i n t e r f a c i a l plane. This r e s u l t i s q u a l i t a t i v e l y i n agreement with the p r e d i c t i o n s o f f i r s t order p e r t u r b a t i o n theory f o r a n i s o t r o p i c overlap forces (14). A p e r t u r b a t i o n treatment using the f u l l atom-atom p o t e n t i a l i s under way. The p a i r p o t e n t i a l i s c u r r e n t l y being modified by the i n c l u s i o n of a quadrupole-quadrupole p o t e n t i a l , and the e f f e c t of s u r f a c e area on s u r f a c e thickness found i n (7) f o r monatomic molecules i s being i n v e s t i g a t e d . 2

Q

2

2

Acknowledgment It i s a pleasure to thank the N a t i o n a l Science Foundation and the Petroleum Research Fund, administered by the American Chemical S o c i e t y , f o r grants i n support of t h i s work, and S o h a i l Murad f o r a copy of h i s s i m u l a t i o n program f o r hydrogen c h l o r i d e .

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

7. THOMPSON AND GUBBiNs

Liquid-Vapor Surface of Molecular85Fluids

Abstract An application of the molecular dynamics method to simulate the liquid-vapor surface of molecular fluids is described. A predictor-corrector algorithm is used to solve the equations of translational and rotational motion, where the orientations of molecules are expressed in quaternions. The method is illustrat­ ed with simulations of 216 homonuclear (N and Cl ) diatomic molecules. Properties calculated include surface tensions and density-orientation profiles. 2

2

Literature Cited 1. Chapela, G. Α., Saville, G. and Rowlinson, J. S., Faraday Disc. Chem. Soc. (1975) 59, 22. 2. Lee, J. Κ., Barker (1974) 60, 1976. 3. Abraham, F. F., Schreiber, D. E. and Barker, J. Α., J. Chem. Phys. (1975) 62, 1958. 4. Opitz, A. C. L., Phys. Letters A (1974) 47, 439. 5. Liu, K. S., J. Chem. Phys. (1974) 60, 4226. 6. Rao, M. and Levesque, D., J. Chem. Phys. (1976) 65, 3233. 7. Chapela, G. Α., Saville, G., Thompson, S. M. and Rowlinson, J. S., Trans. Far. Soc. II (1977) 73, 1133. 8. Sweet, J. R. and Steele, W. Α., J. Chem. Phys. (1967) 47, 3029. 9. Cheung, P. S. Y. and Powles, J. G., Mol. Phys. (1975) 30, 921. 10. Singer, Κ., Taylor, A. and Singer, J. V. L., Mol. Phys. (1977) 33, 1757. 11. Evans, D. J. and Murad, S., Mol. Phys. (1977) 34, 327. 12. Vargaftik, Ν. Β., Tables on the Thermodphysical Properties of Liquids and Gases, 2nd ed., Halsted Press Div., Wiley, New York (1975). 13. Jasper, J. J., J. Phys. Chem. Ref. Data (1972) 1, 841. 14. Haile, J. M., Gubbins, Κ. E. and Gray, C. G., J. Chem. Phys. (1976) 64, 1852. RECEIVED

August 15, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

8 High Field Conductivity BENSON R. SUNDHEIM Department of Chemistry, New York University, 4 Washington Place, New York, NY 10003

In the computatio transpor propertie molecular dynamics data, both direct and indirect (fluctuation-regression) means have been used. The indirect method deals with systems at equilibrium whereas the direct methods do not. Here we study the transport properties associated with electrical fields by the examination of the steady-state properties of a simulated fused salt exposed to a uniform f i e l d . The latter is large by laboratory standards in order to produce stati s t i c a l l y useful displacements. This means that the upper portion of the linear response is explored in a way that is not readily accomplished in the laboratory. Since large fields imply that there must be substantial heat dissipation, i t is necessary to "thermostat" the system so that, as potential energy is withdrawn from the electric f i e l d , kinetic energy is withdrawn from the system. By monitoring this withdrawal, an alternative determination of the conductance can be made. Several interesting by-products of this computer "experiment" are discussed below, including the rate at which kinetic energy in one degree of freedom is "thermalized" into others, the properties of the fluctuating dipole moment per unit volume and the relation between experiments in the laboratory and the computer results. A technical modification in the means of computations has led to the use of a relatively large (40 A) cell so that the relatively long wave length propagating modes can be explored. The molecular dynamics calculation of electrical conductivity in ionic fluids has been approached in several different ways. The autocorrelation function of the current may be related to the conductivity by 0-8412-0463-2/78/47-086-086$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

8.

SUNDHEIM

High Field

Conductivity

Kubo type relations (1_}. By expansion i t may be separated into terms containing only the autocorrelation function of a given particle and terms contain crosscorrelation functions. The latter can be identified as being the source of deviations from the NernstEinstein equation. (_2) A second, quite novel method, is to treat the electric f i e l d as producing a small perturbation to the particle trajectories (3). The molecular dynamics calculation is carried out twice, once without the f i e l d and once with the f i e l d applied as a step function or as an impulse function. By comparison of the two sets of calculations, i t is possible to obtain an estimate of the electrical conductivity. The accuracy and, indeed, the fundamental justification fcr the method has not yet b e e n well established. Finally, there is Remembering that the motion produced b y an external electrical f i e l d is a small perturbation on the Brownian motion, it can be seen that very high fields can be applied without sensibly altering the gross properties. In computer experiments, millions of volts per centimeter can b e applied without concern for electrode processes and thermostatting can b e supplied to prevent significant temperature changes. Consequently, it is possible to examine the range of applicability of Ohm's Law and the details of the transfer of energy from the f i e l d to one component of the kinetic energy and hence to a l l three degrees of freedom and ultimately into the thermostatic bath. The conductivity can be determined both by determining the rate at which heat is extracted from the system and by determining the particle mobility in the applied f i e l d . It is this high f i e l d method which is the f i r s t subject of this communication. Computational Details For convenience, the potential energy of interaction, aside from the coulombic term, was chosen to be the same for a l l particles, being a rough approximation to that appropriate to molten KC1. Its form was that recommended by Woodcock (!5). The symmetry of the potential plus the absence of ionic polarization terms means that the results are not specific to any real salt but rather pertain only to this model. On the other hand, the main results are meant to refer to the properties associated with dense ionic fluids in general and not to unique properties associated with various anomalies in the

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

88

COMPUTER

MODELING

OF MATTER

potential energy of interaction. The coulombic contribution was evaluated by the Ewald method (5,J5). The direct force contribution was evaluated Tor a pair of particles by using a precalculated force/distance table based on the potential energy function. Pairs with distances (either direct or to the nearest image) greater than 6.6 A were omitted by use of a "link" system (7) as adapted to fused salt systems by Woodcock (8). The applied electrostatic f i e l d was simulated by adding a constant force term (of opposite sign for the two ionic species) to the direct force on each particle. This results in a flow of energy into the system. Compens a t o r thermostatting was applied at each time step (10" sec) by determining the ratio of the net kinetic energy to that required by the nominal temperature and dividing eac of this ratio. The Verle for the integration. Autocorrelation functions were computed by the Fourier transform method.(10). The rescaling of the velocities in order to maintain the desired rms velocity in each cartesian coordinate is an approximation to thermostating.A . better procedure would be to scale in such a way as to maintain a Boltzman distribution. The linear scale factor utilized here represents the leading term in a power series expansion in such a Boltzman rescaling. Noting that the average correction coefficient is less than 0.998 even for very high f i e l d s , i t may be readily seen that the relative deviation between the distribution achieved and a Boltzman distribution is quite small and unimportant in these studies. The rescaling plays no role in the momentum balance so that the system may be described as a constant temperature, constant potential gradient, constant momentum, constant density ensemble. (The density is constant over a spatial "graining" of the orjjer of the cell volume, here approximately 40x40x40 A, and the temperature is constant over the temporal graining of 10 sec.) Experimental measurements of electrical conductivity in real systems ignore heating effects wherever possible, or remove them by extrapolation to zero applied f i e l d . In our computations a similar extrapolation to zero applied f i e l d was made. Results We turn now to an examination of the results the computations which illustrate the method.

of

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

SUNDHEIM

High Field

Conductivity

The average current flow over 500 steps in systems which had previously come to a steady state was used to calculate the electrical conductivity. The results are summarized in F i g . l . A slight apparent positive slope to the least squares line i s , in fact, well within the 95% confidence level of zero slope (dashed line) and we conclude that up to 15 megavolts per cm the isothermal conductivity is constant for this model. (Further decreases in the uncertainty of the slope, obtainable by longer runs and/or more points, did not seem to be especially valuable.) The underlying reason for the constancy of the conductivity i s , of course, the fact that the external force is only a tiny fraction of the rms fluctuating force experienced by a particle. There are two other means of obtaining the elect r i c a l conductivity fro amount of heat extracted from the system at each step was recorded and divided by the square of the current. The resulting quantity is also the conductivity. Values obtained in this way are also shown in F i g . l (+'s) . Finally the power spectrum of the electrical current in a f i e l d free system was computed and the autocorrelation function derived therefrom used to obtain the electrical conductivity via the well-known relationship (1,2). The value so obtained is entered on F i g . l as an open c i r c l e . AH of these methods are, in principle, equivalent. The highest accuracy is attached to the direct measurement of the mean current. The diffusion coefficient can be obtained by the limiting slope of the graph of the mean square displacement per particle versus time. Alternatively, the velocity autocorrelation function may be utilized for the calculation (1,2).The s t a t i s t i c a l reliability of the velocity autocorrelation function is quite high for this large system since i t represents the mean of the autocorrelation function for 1728 particles over several sets of 500 steps. Comments There are two points peculiar to the computation of electrical conductivity that should be discussed here. One has to do with the mode of the electrostatic current waves and the other with dielectric shielding. Both transverse and longitudinal current oscillations occur in the molecular dynamics computations (6). The periodic boundary conditions mean that the

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

90

COMPUTER

MODELING

10

I10

6

O F MATTER

20

V/cm )

Figure 1. Computed conductivity as a function of applied field. The solid line represents the least squares fit to the high-field method points and the dashed line the mean value of these points. The open circle is the zero field point obtained by the current correlation function method. The crosses are the points obtained by the heat dissipation method.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

SUNDHEIM

High Field

Conductivity

current will be divergenceless for wavelengths equal to or greater than the dimensions of the elementary c e l l . When the DC current is obtained by averaging over the c e l l , longitudinal currents are eliminated. (The zero k vector wave does not occur since the periodic boundary conditions imply a "toroidal" space which is also divergenceless at + infinity and hence has no planes of charge accumulation.) These calculations therefore refer to a zero k vector transverse wave, which is the quantity measured on a real salt with reversible electrodes and direct current. The dielectric properties are c l a s s i c a l l y pictured in terms of "free" and "bound" charges associated with an electrical f l u i d . In a fused salt system, charge is carried by both types of particles and, for non-polar particles which can be bound. Nevertheless, i t is possible to polarize the system of ions by inducing a charge separation. In the simplest terms, an external field biases the local distribution functions, giving rise to a reaction f i e l d which modifies the mean f l u c t uating force on each particle. We may cast this in the conventional form by the following argument. The linear expression for the current-voltage relation is (11) J =E/at) In an isotropic f l u i d *~ andX are scalars. polarization per elementary cell of edge S is

The

P= 2 z . r.-n.S I I I I where n is the (integral) number of displacements through the unit cell wall experienced by a particle which was originally in the cell centered at (0,0,0). Then fc

PP/H) = £ z . v . - Sg>n./>t) c Since ^z^v- = J, 1

the net current observed by averaging over the cell and the "mobile charge current", j may be written as j = S£"0n./»t)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

92

COMPUTER

MODELING

O F MATTER

Then J = j + PP/At) Thus, pP/^t) plays the role of the "bound charge current". Of course, in the steady state J = j . By comparison, since j = «"E p -yi by setting E = D -

4 P7r

D =£E so that
In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

8.

SUNDHEIM

High Field

Conductivity References

1.

a. G r e e n , M . S . , J. Chem. P h y s . , 22, 398 (1954). b. Kubo, R . , J. Phys. Soc. Japan, 12, 570 (1957). c . Kubo, R., Rep. Prog. P h y s . , 29, 255 (1966). d. Zwanzig, R . , Ann. Rev. Phys. Chem., 16, 67 (1967). 2. Hansen, J. P . and McDonald, E.L., J. Chem. P h y s . , 62, 4581 (1975). 3. ( G i c o t t i , G. and J a c u c c i , G., Phys. Rev. Lett., 31, 206 (1973). 4. c f . the a p p l i c a t i o n of t h i s method to viscosity by a. A s h u r s t , W. T. and Wainwright, T. E., J. Chem. P h y s . , 5 3 , 3813 (1970). b. G o s l i n g , E. M., S i n g e r , I . R. and S i n g e r , K., M o l . P h y s . , 26, 1475 (1973). 5. e . g . , Woodcock, E S a l t Chemistry, V o l . 3 , ed. B r a u n s t e i n , J. Mamontov, G. and Smith, G. P., Plenum P r e s s , New 6. Hansen, J.P. and McDonald, E . L . , Theory of Simple L i q u i d s , Acad. P r e s s , New York (1976). 7. a. S c h o f i e l d , P., Comput. Phys. Comm., 5, 17 1973). b. Quentrec, B. and B r o t , C., J. Comput. P h y s . , 13, 430 (1973). 8. Woodcock, L.E., p r i v a t e communication. 9. V e r l e t , L., Phys. R e v . , 159, 98 (1967). 10. Futrelle R . P . and McGinty, D. J., Chem. Phys. Lett., 12, 785 (1971). 11. Landau, L . D . and Lifshitz E . M . ,Vol.5, p 391 ff., Pergamon P r e s s , New York (1958). 12. Valuable conversations with D r . L.E. Woodcock (Cambridge), grants of computer time from the Courant Institute, N . Y . U . under D.O.E grant E(11-1)-3077 and support from the National Science Foundation are gratefully acknowledged. RECEIVED

August

1 5 , 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9 Computer Simulation of Collective Modes in Solids M. L. KLEIN Chemistry Division, National Research Council of Canada, Ottawa, Canada K1A 0R6

The molecular dynamic c a l c u l a t e the timedependen This paper reviews the a p p l i c a t i o n o f this computer s i m u l a t i o n technique t o the c a l c u l a t i o n o f the normal modes o f v i b r a t i o n o f s o l i d s and t h e i r l i f e t i m e s . Examples will be r e p r e s e n t a t i v e o f most c l a s s e s o f solid found i n nature, namely the i n e r t gases, i o n i c c r y s t a l s , metals, a l l o y s , and molecular s o l i d s . Normal modes o f v i b r a t i o n o f s o l i d s are r o u t i n e l y s t u d i e d experimentally u s i n g i n e l a s t i c s c a t t e r i n g o f neutrons or photons ( 1 ) . A l s o the theory o f the dynamics o f c r y s t a l s is now w e l l developed and even anharmonic e f f e c t s are u s u a l l y i n c o r p o r a t e d i n t o t h e o r e t i c a l models ( 1 ) . T h i s being the case one may wonder why computer s i m u l a t i o n is a t all necessary? W e l l , neutron s c a t t e r i n g i s r a t h e r c o s t l y so that although t h i s technique is in p r i n c i p a l very powerful it is l i m i t e d i n t h i s sense and light s c a t t e r i n g i s mostly r e s t r i c t e d t o small wave v e c t o r s . The s i m u l a t i o n method, on the other hand, i s r e a l l y a compliment t o the r e a l experiments s i n c e i t allows one t o compare approximate t h e o r e t i c a l approaches t o the c r y s t a l dynamics with exact r e s u l t s for a given i n t e r m o l e c u l a r f o r c e model, subject o n l y to the l i m i t a t i o n that c l a s s i c a l s t a t i s t i c a l mechanics a p p l i e s . F o r t u ­ n a t e l y , f o r most s o l i d s , t h i s l a t t e r r e s t r i c t i o n i s not a severe l i m i t a t i o n . Moreover, i n the computer s i m u l a t i o n s t h e r e i s an unambiguous d i s t i n c t i o n between coherent and incoherent s c a t t e r i n g , and between one-phonon and multiphonon e f f e c t s , a s i t u a t i o n which does not always p e r t a i n to r e a l experiments. We have used the molecular dynamics (MD) method o r i g i n a l l y pioneered by Alder and Wainwright (2) f o r hard spheres and by Rahman ( 3 ) f o r continuous potentials. The c o l l e c t i v e modes (phonons) we wish t o study are r e l a t e d t o the spectrum o f d e n s i t y f l u c t u a t i o n s i n the s o l i d . Such motions are obtained by i n t e g r a t i n g the c l a s s i c a l equations of motion (using a time step o f 10"~ to 1 0 ~ sees) f o r a model system composed o f approximately 1 0 p a r t i c l e s , i n t e r a c t i n g v i a some assumed f o r c e law, with the p e r i o d i c boundary c o n d i t i o n being used t o simulate an i n f i n i t e system. These boundary 11+

1 5

3

0-8412-0463-2/78/47-086-094$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9.

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Collective Modes in Solids

95

c o n d i t i o n s "quantize" the allowed wave v e c t o r s t h a t can "be studied. The d e t a i l s o f the phase space t r a j e c t o r i e s are stored so that at a l a t e r time s t a t i s t i c a l averages can be performed. This i n essence i s the method. I t has been widely discussed i n the l i t e r a t u r e and w i l l not be d e s c r i b e d i n d e t a i l here. The

Dynamical S t r u c t u r e Factor S(Q,a))

I t i s w e l l known that apart from d u l l f a c t o r s the coherent s c a t t e r i n g o f neutrons from l i q u i d s and s o l i d s i s determined by the Van Hove f u n c t i o n (dynamical s t r u c t u r e f a c t o r )

S(Q,(o) =

dte

i a ) t

F(Q,t)

where F(Q,t) i s the intermediat t u r n i s r e l a t e d t o the time c o r r e l a t i o n f u n c t i o n o f the operator

p (t) = I y i=l

e ^

Q

{

t

density

)

where N i s the number o f p a r t i c l e s i n the system whose p o s i t i o n at time t are given by r ( t ) . The momentum and energy t r a n s f e r r e d to the system i n the s c a t t e r i n g process are Q and w r e s p e c t i v e l y . In d e t a i l F(Q,t) = ( l / N ) < p ( t ) p Q

(0)> ,

where the angular bracket denotes a s t a t i s t i c a l average which w i l l be evaluated by the c l a s s i c a l molecular dynamics method (MD). The f i r s t c a l c u l a t i o n s o f t h i s type were c a r r i e d out f o r l i q u i d r a r e gases near the t r i p l e p o i n t (h) . Since then a v a r i e t y o f f l u i d s have been s t u d i e d i n c l u d i n g l i q u i d metals (5.), l i q u i d n i t r o g e n (6), the c l a s s i c a l one-component plasma (7.)» molten s a l t s (8) and water (9.). Because l i q u i d s are i s o t r o p i c S(g,a)) depends only on |g| whereas f o r s o l i d s t h i s i s not t r u e . A f u r t h e r d i s t i n c t i o n a r i s e s because f o r many s o l i d s under a wide v a r i e t y o f s t a t e c o n d i t i o n s the c o n s t i t u e n t p a r t i c l e s execute small amplitude v i b r a t i o n s about w e l l defined e q u i l i b r i u m p o s i ­ t i o n s (noteworthy exceptions are p l a s t i c c r y s t a l s ) . If this s i t u a t i o n p e r t a i n s i t i s p o s s i b l e t o express r ^ ( t ) = R$+u£(t) the instantaneous p o s i t i o n o f p a r t i c l e £ i n terms o f i t s mean p o s i ­ tion and i t s time dependent displacement uj^(t) . I f U£ i s i n some sense small with respect t o R^ one can develop F(§,t) as a power s e r i e s i n §*u. This i s the s o - c a l l e d phonon expansion (10)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

96

COMPUTER

F(Q,t) = Fo + F + F . x

n t

MODELING

OF MATTER

+F + 2

where the successive terms d e s c r i b e the e l a s t i c (zero-phonon) s c a t t e r i n g one-phonon i n e l a s t i c s c a t t e r i n g , the i n t e r f e r e n c e between one and two phonon s c a t t e r i n g , and the two-phonon s c a t t e r i n g , e t c . For a c l a s s i c a l s o l i d these i n d i v i d u a l c o r r e ­ l a t i o n f u n c t i o n s can be evaluated u s i n g the MD method as can the f u l l F(Q,t). From a t e c h n i c a l p o i n t o f view one can proceed as f o l l o w s . One can evaluate F(Q,t) f o r a given Q and then take the F o u r i e r transform. A l t e r n a t i v e l y , one can proceed v i a the F o u r i e r Laplace transform o f the d e n s i t y operator PQ(CO) u s i n g the formula T

T

N S(Q,o)) = l i m j e T

i u ) t

p (t)dt j ?

e~

i ( A ) t

' p _ ( t ' )dt ?

J^

3

~* °

= lim|p

(w)\ / 2

X

Both methods have been used i n the l i t e r a t u r e (11). Other methods i n v o l v i n g p e r t u r b a t i o n s o f t r a j e c t o r i e s have a l s o been used t o study c o l l e c t i v e modes i n s o l i d s (12). The one-phonon approximation t o the dynamical s t r u c t u r e f a c t o r S i can be w r i t t e n (13) N S!(<3,u)) = lim|p (a)) |

2

Q

where p (t) = e~ Q

W

1

I e ®"?* Q.u (t) 1=1 £

and W * i|Q|2

< u

2>

i s the exponent o f the Debye-Waller f a c t o r . Since t h i s i s the q u a n t i t y most u s u a l l y c a l c u l a t e d by approx­ imate t h e o r e t i c a l models (10) i t i s o f c o n s i d e r a b l e i n t e r e s t t o evaluate S I ( Q , O J ) and t o compare t h i s with the f u l l S(Q,o)) f o r a given model. In the f o l l o w i n g s e c t i o n s we consider i n t u r n the a p p l i c a t i o n o f the MD technique t o Rare Gas S o l i d s (13.), A l k a l i Halides (lh), Metals (l5,l6) and A l l o y s (1T_) and Molecular C r y s t a l s (l8,19).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9.

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97

Collective Modes in Solids

Rare Gas S o l i d s The r a r e gas s o l i d s u s u a l l y c r y s t a l l i n e i n a fee s t r u c t u r e . The interatomic f o r c e s are predominantly p a i r w i s e a d d i t i v e but there i s a c o n s i d e r a b l e body o f evidence t o suggest that q u a l i t a ­ t i v e models r e q u i r e the presence o f 3-body c o n t r i b u t i o n s (20). Nevertheless the Mie-Lennard-Jones (12-6) p o t e n t i a l provides a good e f f e c t i v e model with which t o study the r a r e gases.

v(R) = M ( o 7 R ) - ( a / R ) ] 1 2

6

3

A convenient u n i t o f temperature i s T*=kgT/e and d e n s i t y p* = pa , where f o r s o l i d Ar e/kg - 120 K and a - 3.^0 A. Extensive c a l c u ­ l a t i o n s o f S(Q,o)) f o r the (12-6) p o t e n t i a l were c a r r i e d out by Hansen and K l e i n (13). They c a r r i e d out systematic s t u d i e s o f the temperature and d e n s i t y dependence o f the response f u n c t i o n T y p i c a l s p e c t r a are show corresponds t o 18 cm"" upper h a l f o f the f i g u r e corresponds t o approximately one h a l f the m e l t i n g p o i n t o f s o l i d Ar while the lower curve i s c l o s e t o the t r i p l e p o i n t . The f u l l l i n e s are the X-point response which i n c l u d e s both l o n g i t u d i n a l and t r a v e r s e phonons f o r the zone boundary <001> d i r e c t i o n . The dashed dot curve shows the l o n g i ­ t u d i n a l response alone. In general the high frequency phonons were very broad at high temperatures and t h i s e x p l a i n s why they have proved d i f f i c u l t t o measure experimentally. One novel feature o f the simulations (13) was the presence of a c e n t r a l Rayleigh peak f o r the smallest wave v e c t o r Q=(27r/a) (1/8,0,0) studied a t high temperatures i n the system o f 20hQ particles. The width o f t h i s peak i s governed by the thermal d i f f u s i v i t y and provides a crude estimate o f the thermal conduc­ t i v i t y since the s p e c i f i c heat i s known from other c a l c u l a t i o n s . Alkali

Halides

The f i r s t MD study o f a l k a l i h a l i d e s u t i l i s e d the r i g i d i o n model. This venerable o l d model c o n s i s t s o f point charges plus Born-Mayer r e p u l s i o n s and d i s p e r s i o n f o r c e a t t r a c t i o n s . The l i m i t a t i o n s o f t h i s model are w e l l known t o be due p r i n c i p a l l y t o neglect o f p o l a r i z a t i o n e f f e c t s . The subject o f the interatomic f o r c e s has been reviewed r e c e n t l y by Sangster and Dixon (21). J a c u c c i et a l . (lU) studied NaCl a t s e v e r a l temperatures u s i n g the MD method. A system o f 2l6 ions was used and the Coulombic i n t e r a c t i o n s were handled by an Ewald method (21). Results f o r the smallest wave v e c t o r studied a t 302 K are shown i n Figure 2. The i n s e r t f i g u r e shows the L0 ( l o n g i t u d i n a l Optic) phonon where­ as the main peak corresponds t o the LA ( L o n g i t u d i n a l A c c o u s t i c ) mode. Figure 3 shows the same phonon c l o s e t o the melting point of the model (somewhat higher than that o f the r e a l s o l i d ) . We see that the phonons have broadened and a c e n t r a l peak appears t o

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

98

COMPUTER

MODELING

OF

MATTER

Figure 1. The upper curves show S(Q,(o) data for a 2048 particle Lennard-Jones (12-6) system after 11,000 time steps (of about 10' sees for argon). The lower curves are for the same Q values but at a higher temperature at lower density after 30,000 time steps. 14

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

KLEIN

Collective Modes in Solids

•

Q = (2TT/3I)( 1,0,0)

1 1 1

6 -

T =

302 K

j

I II II

.02-

II

4

•

H

h

A /

^ 3

!! 1 1 1| 1 1 • i i *

2 —

/ 5

\

.01-

\^ 6

j! i i

\

0

1

2

I

I

3 CJ in 1 0

4 13

rad /sec

Figure 2. S(Q,w) for the smallest accessible wave vector of a system of 216 ions (Na*Cl) at 302 K. The dots show the LA response. The inset shows the high frequency LO response on the same relative scale (full line) and the charge density response (dashed line) on an arbitrary relative scale.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

100

Q = (2 TT / 3a) (1,0,0)

•

T = 1153 K

• • -

.03-

5

4 • •

I •

1

I 1

H •

•

•

I—

1

1 1

• •

0

6

1

I

M

1 1

• •

r s

2

CO in 1 0

i

3 13

i 4

5

rad / sec

Figure 3. S(Q o>) for the smallest accessible wave vector of a system of 216 ions (Na*Cl~) at 1153 K. The dots shoto the LA response. The inset shows the high frequency LO response on the same relative scale (full line) and the charge density response (dashed line) on an arbitrary relative scale. y

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9.

KLEIN

Collective

Modes in Solids

101

be present a l s o . The LA mode appears t o be i n reasonable agree­ ment with experimental data where comparison can be made but t h e L0 mode i s much too high i n frequency due t o neglect o f p o l a r i z a ­ t i o n e f f e c t s (22). In subsequent c a l c u l a t i o n s such e f f e c t s were incorporated v i a a s h e l l model and a dramatic lowering o f the LO peak p o s i t i o n was found (23) see F i g u r e k. F i n a l l y , the i n t e r ­ ference e f f e c t between the one phonon peak and the multiphonon background was a l s o studied and s e v e r a l novel e f f e c t s were predicted. Figure 5 shows the Q dependence o f S(Q,u)) f o r phonons that would be equivalent i n the harmonic approximation or i n the one phonon approximation apart from t h e d u l l f a c t o r s of Q^e"^. C l e a r l y important c o n t r i b u t i o n s a r i s e from the higher order terms i n the phonon expansion. Metals and A l l o y s The a l k a l i metals derivation of e f f e c t i v p r i n c i p a l s (2k). The method has a l s o been a p p l i e d t o d e r i v e p o t e n t i a l s f o r A l . These p o t e n t i a l s have been used t o c a l c u l a t e the dynamical s t r u c t u r e f a c t o r s S(Q,co) f o r bcc K at temperatures corresponding t o approximately one h a l f and nine-tenths melting (15). A t y p i c a l phonon i s shown i n F i g u r e 6. The temperature dependence o f both the s h i f t and width o f t h i s peak agree w e l l with experimental data. F i g u r e 7 i l l u s t r a t e s the i n t e r f e r e n c e e f f e c t between the one phonon peak and the multiphonon background rather n i c e l y . The peaks would be i d e n t i c a l i n shape i n t h e one phonon approximation. However, the i n t e n s i t y d i f f e r e n c e s on the l e f t hand side o f the phonon peak and t h e s h i f t i n peak p o s i t i o n are due t o the c o u p l i n g o f the one phonon peak t o the background. D e t a i l e d comparisons with l a t t i c e dynamical c a l c u l a t i o n s have been c a r r i e d out (25). Some S(Q,u)) s p e c t r a f o r s o l i d fee A l c l o s e t o m e l t i n g are shown i n Figure 8 by t h e dotted curves there appears t o be broadly s i m i l a r t o spectra f o r fee Ar s u i t a b l y s c a l e d (l6). Some i n t e r e s t i n g i n t e r f e r e n c e e f f e c t s have been p r e d i c t e d (l6) t o be observable i n A l . The main i n t e r e s t i n these metals has however, centred on the study o f d e f e c t s (26) and t h e i r m i g r a t i o n by MD methods but such studies go beyond t h e scope o f t h i s a r t i c l e . F i n a l l y , the method has been extended t o study l o c a l i s e d modes o f v i b r a t i o n a s s o c i a t e d with l i g h t i m p u r i t i e s i n a l l o y s . The system K 2 9 K D 7 1 has been e x t e n s i v e l y studied (17) and d e t a i l e d r e s u l t s w i l l be published i n due course. The c o r r e l a t i o n with a v a i l a b l e experimental information i s e x c e l l e n t . The time e v o l u t i o n o f a quenched b i n a r y a l l o y has a l s o been s t u d i e d u s i n g MD (27). Molecular C r y s t a l s The f i n a l c l a s s o f c r y s t a l s s t u d i e d t o date are molecular crystals. One f e a t u r e that i s o f t e n found i s t h a t such c r y s t a l s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

102

COMPUTER MODELING OF MATTER

Q = (2TT/3a)(l,0,0) T = 302 K

o • SHEL —I

h

*

^oOOOO°

C

_L 5

6

7

CO in 10 rad/sec U

Figure 4. Effect of polarization on the LO mode of smallest wave vector studied. The open circles are for the rigid ion model while the full circles are for the shell model.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9. KLEIN

Collective Modes in Solids

103

Figure 5. The interference effect in S(Q,) for the indicated Q value, and shows how the interference effect transfers intensity from the left hand side of the phonon peak to the right hand side. The dashed-dot curve shows a similar effect but for a larger Q value. Here the situation is complicated somewhat by the multiphonon background. y

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

104

COMPUTER

1.4

1.8

MODELING O F M A T T E R

2.2

cu/27T(THz) Physical Review

Figure 6. S(Q,u)) for solid K. The solid line corresponds to approximately one half of the melting temperature, the dashed and dotted curves are two independent calculations at about nine-tenths melting. The system consisted of 432 atoms interacting via the potential from Kef. 24 truncated after eight shells of neighbors. The inset shows the temperature variation of the peak position and width compared with experiment (error bars) and lattice dynamical calculations (25).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9. KLEIN

Collective Modes in Solids

I

I

.3

1.0

1

105

I

^ T"

1.5 2.0 2.5 CJ/27T(THz)

I

L_

3.0 3.5

Figure 7. The interference effect in S(Q, ) of solid K for the indicated Q* = QSL/2TT values. The lower curves correspond to equivalent phonons either side of the Bragg vector Q* = (2,2,0) while the upper curves refer to similar phonons for either side of the Bragg vector Q* = (3,3,0). W/

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

106

COMPUTER MODELING OF MATTER

(1.0,0)

(0.75,0,0)

f I l I A*

3(0.5,0,0)

Physical Review

Figure 8. S(Q o>) data for longitudinal phonons propagating in the <100> direction of fee Al at 800 K (dotted curves). The unit of frequency is 85 cm' , 2.55 THz, or 10.6 meV. The dashed curves are similar data for rare gas solids arbitrarily scaled (16). y

1

-

(0,0,5)

3 to 0

2

-

4

6

Journal of Chemical Physics

Figure 9. The full curves are the MD S(Q,) data for longitudinal phonons propagating along the c-axis of hep solid fi-Ng. Curves labeled QH and SCHA refer to lattice dynamical theories (19). The unit of energy is 16.5 cm' , 0.50 THz, or 2.05 meV. 1

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9. KLEIN

Collective Modes in Solids

107

Journal of Chemical Physics

Figure 10. The full curves are the MD S(Q,w) data for longitudinal phonons propagating in the basal plane of hep The dashed dot curve shows the center of mass spectrum. The remnant of the Rayleigh peak is to be noted for the smallest Q vector (19).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

108

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MODELING OF MATTER

undergo s t r u c t u r a l phase t r a n s i t i o n i n the s o l i d that are a s s o c i a t e d with d i f f e r i n g degrees o f r o t a t i o n a l freedom of the molecules. The prototype molecular s o l i d i s s o l i d Nitrogen. At low temperatures i t c r y s t a l l i n e s i n the cubic Pa3 s t r u c t u r e (exphase) with four molecules i n the u n i t c e l l . When t h i s s o l i d i s heated i t transforms i n t o the hep r o t a t i o n a l d i s o r d e r e d ( r o t a ­ t i o n a l d i f f u s i o n or p l a s t i c ) 3-phase. At high pressures and low temperatures yet another phase (y) occurs that w i l l not concern us here. The nature of the phonon e x c i t a t i o n s i n the a-phase was studied with MD by Weis and K l e i n (l8) u s i n g an atom-atom (12-6) p o t e n t i a l model. F a i r agreement with experiment was obtained at low temperatures but the s o l i d was found to transform i n t o the disordered 3-phase too r e a d i l y since the model molecule d i d not have a quadrupole moment. The hep 3-phase was a l s o s t u d i e d (19) and s e l e c t e d S(Q,GO) s p e c t r a are shown i n F i g u r e s 9 and 10 f o r l o n g i t u d i n a l phonons propagating along the c - a x i s and i n the basal plane r e s p e c t i v e l y are the enormous change e x c i t a t i o n changes from phonon l i k e motion to predominantly r o t a t i o n a l e x c i t a t i o n . The model appears to c o r r e l a t e w e l l with experimental (19) data. The most s i g n i f i c a n t f i n d i n g o f the c a l c u l a t i o n s was the importance of r o t a t i o n a l - t r a n s l a t i o n a l coupling i n lowering the t r a n s v e r s e shear mode frequencies i n the disordered 3-phase. This probably accounts f o r the low shear constants observed i n a l l p l a s t i c c r y s t a l s . Summary The MD technique has now been used to explore the nature o f S(Q,a)) i n a wide v a r i e t y of c r y s t a l s and important new i n s i g h t s have been gained. In the f u t u r e the method w i l l l i k e l y be extended to hydrogen bonded s o l i d s and other more complicated molecular s o l i d s , polymers, s u p e r i o n i c conductors (28), l i q u i d c r y s t a l s , and two-dimensional f i l m s (2£). E q u a l l y novel and important r e s u l t s can be expected there a l s o . Acknowledgements It i s a pleasure to acknowledge the f r u i t f u l l c o l l a b o r a t i o n s and f r i e n d s h i p s I have enjoyed with Jean P i e r r e Hansen, G i a n i J a c u c c i , Ian McDonald and Jean-Jaques Weis that are embodied i n this article. The i n v a l u a b l e advice of Dom Levesque and Aneesur Rahman has a l s o been generously given wherever sought a f t e r and was much a p p r e c i a t e d . Literature Cited 1.

Horton, G.K. and Maradudin, A.A., e d i t o r s , "Dynamical P r o p e r t i e s o f S o l i d s " V o l s . I and I I , N. Holland, Amsterdam

(1974,1975).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

9. KLEIN 2.

Collective Modes in Solids

109

A l d e r , B.J. and Wainwright, T.E., J. Chem. Phys. 31,

(1959),

459.

3.

Rahman, A., Phys. Rev. ( 1 9 6 4 ) 136, A 4 0 5 .

4.

Levesque, D . , Verlet, L . and (1974),

A7,

Kürkijarvi,

J.,

Phys. Rev.

1690.

5.

Rahman, A., Phys. Rev.

6.

Weis, J.J. and Levesque, D . , Phys. Rev. ( 1 9 7 6 ) A13, 4 5 0 .

7.

Hansen, Lett.

8.

J.P.,

(1974),

Lett.

(1974),

32, 52.

P o l l a c k , E.L. and McDonald, I.R., Phys. Rev. 32, 277.

Hansen, J.P. and McDonald, I.R., Phys. Rev.

(1974)

All,

(1974)

A 1 0 , 368.

2111.

9.

Rahman, A . and

Stillirger

F.H.,

Phys. Rev.

10.

Maradudin, A . A . and Fein, A.E., Phys. Rev.

( 1 9 6 4 ) 128, 2589.

11.

Hansen, J.P. and Klein, M.L., J. Phys. L e t t .

( 1 9 7 4 ) 35,

L-29.

12.

Dickey, J . M . and P a s k i n A., Phys. Rev. (1969) 188, 1407.

13.

Hansen, J.P. and Klein, M.L., Phys. Rev. (1976) B 1 3 , 878.

14.

Jacucci, (1975)

15.

G.,

36,

Klein

M . L . and McDonald, I.R., J. Phys.

Lett.

L-97.

Hansen, J.P. and

Klein,

M.L.,

Solid.

S t . Comm. ( 1 9 7 6 ) 20,

771.

16.

J a c u c c i , G . and Klein, M.L., Phys. Rev. (1977) B16, 1322.

17.

Jacucci, (1977)

G.,

24,

Klein,

M . L . and T a y l o r ,

R.,

Solid.

S t . Comm.

685.

18.

Weis, J.J. and Klein, M.L., J. Chem. Phys. ( 1 9 7 5 )

63,

19.

Klein,

20.

Klein M . L . and Venables, J.A., e d i t o r s , "Rare Gas Vol. I and II, Academic P r e s s , London (1975,1977).

21.

Sangster,

2869.

M . L . and Weis, J.J., J. Chem. Phys. ( 1 9 7 7 ) 67, 217.

M.J.L.

Solids"

and D i x o n , M., Adv. in Phys. ( 1 9 7 6 ) 25,

247.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

110

COMPUTER MODELING OF MATTER

22.

Cowley, E.R., J a c u c c i , G., K l e i n , M.L. and McDonald, I.R., Phys. Rev. (1976) B14, 1 7 5 8 .

23.

J a c u c c i , G., McDonald, (1976) A13, 1 5 8 1 .

24.

Dagens, L., R a s a l t , M. and T a y l o r , R., Phys. Rev. (1975) B11, 2726.

25.

Glyde, H.R., Hansen, J.P. and K l e i n , M.L., Phys. Rev. (1977) B16, 3476.

26.

Da Fano, A. and J a c u c c i , G., Phys. Rev. L e t t . (1977) 3 9 , 950.

27.

Sur, A., Lebowitz, J.L., Marro, J. and Kalos, M.H., Rev. (1977) B15, 3014

28.

Rahman, A., J. Chem

29.

Hanson, F.E., Mandell, M.J. and McTague, J.P., J. Phys. (1977) c-4, 7 6 .

RECEIVED August 15,

I.R. and Rahman, A., Phys. Rev.

1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Phys.

10

Computer Simulations of the Melting and Freezing of Simple Systems Using an Array Processor G. CHESTER, R. GANN, R. GALLAGHER, and A. GRIMISON Laboratory of Solid State Physics and Office of Computer Services, Cornell University, Ithaca, NY 14853

The aim of t h i s work is to study, at a microscopic l e v e l , pre­ cursor phenomenon to f r e e z i n g and m e l t i n g . At the present time the studies are being c a r r i e d out on s i n g l e phases, c r y s t a l and fluid (1,2,3). In the near future we expect to extend some o f our work to two phases i n equilibria. We hope that these studies will u l i t m a t e l y lead to an understanding of how the long range s p a t i a l order o f a c r y s t a l i s destroyed by m e l t i n g and how the same order is built up from the c h a o t i c f l u i d phase. These questions can best be studied initially i n the simplest systems. For t h i s r e a ­ son we have been studying c e n t r a l force classical systems i n both two and three dimensions. The d i f f e r e n c e s i n dimensions may lead to important i n s i g h t s about the o r d e r i n g phenomenon. The p r o p e r t i e s we plan t o c a l c u l a t e by Metropolis (4) Monte Carlo methods are the s t r u c t u r e f a c t o r s S(k) a t d e n s i t i e s and temperatures near the f r e e z i n g l i n e , the root mean square d i s ­ placement o f the p a r t i c l e s from t h e i r l a t t i c e s i t e s near m e l t i n g , the angular c o r r e l a t i o n s between d i s t a n t p a i r s o f p a r t i c l e s i n the c r y s t a l phase, and near-neighbour c o r r e l a t i o n s i n the c r y s t a l phase. Several o f these p r o p e r t i e s r e q u i r e very long simulations to achieve good e q u i l i b r i u m v a l u e s . We have t h e r e f o r e f r e q u e n t l y made runs a t l e a s t one order of magnitude l a r g e r than i s normal i n s i n g l e phase studies o f c e n t r a l p o t e n t i a l systems. Determining the s i z e dependence involves a s i m i l a r amount o f computing. In a d d i t i o n , to extend these s t u d i e s , at l e a s t i n two dimensions, to two phase simulations r e q u i r e s between 1000 and 10,000 p a r t i c l e s and we estimate that runs equivalent to between 3 and 30 hours o f CDC 7600 computer time will have to be made. A f u r t h e r considera­ t i o n i s that we p l a n to complement the Monte Carlo work with molecular dynamics s i m u l a t i o n s . These again r e q u i r e very long

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runs i n order to simulate phenomena which occur comparatively slowly i n r e a l time. For example the break up o f d i s l o c a t i o n p a i r s i n two dimensions may only occur on time s c a l e s o f at l e a s t 10^ conventional time steps o f molecular dynamics. Previous s o l u t i o n s to the problem o f such l a r g e - s c a l e compu­ t a t i o n s have mainly involved large powerful shared machines (IBM 370, CDC 7600, CDC S t a r , CRAY 1) or dedicated minicomputers (Prime 400, H a r r i s / 4 , PDP 11/70, e t c . ) . Each o f these s o l u t i o n s has i t s dedicated advocates, and there are h i g h l y favorable features i n each case. The large machine o f f e r s large memory, d i s k storage and powerful p e r i p h e r a l s , very extensive software, and a f a s t computation time. The large machines l i s t e d above are capable of o p e r a t i n g i n the range o f 2 to 100 megaflops ( m i l l i o n s o f f l o a t i n g p o i n t operations per second). The g r e a t e s t disadvantage i s o f t e n the high cost o f u s i n g such a system, unless i t i s h i g h l y s u b s i d i z e d and then somewhat r e s t r i c t e d i n a c c e s s . On a very h e a v i l can a l s o be a problem, system tends to be "Ho much w i l l i t c o s t ? " . The greatest advantage o f the dedicated mini-computer i s probably the low cost (50, a l l i e d with the convenience o f c o n t r o l of the r e s o u r c e s . This must be o f f s e t by the more l i m i t e d memory and storage a v a i l a b l e , the l i m i t e d p r e c i s i o n o f the hardware i n some cases, and the l e s s extensive software l i b r a r i e s a v a i l a b l e . However, the p r i n c i p a l l i m i t a t i o n f o r very l a r g e - s c a l e computa­ t i o n s stems from the lower computational speed (around 0.5 mega­ f l o p s or l e s s ) so that the users concern i s o f t e n "How long w i l l i t take?". Recently r e l a t i v e l y cheap s p e c i a l purpose "attached proces­ s o r s " have become a v a i l a b l e , and t h i s paper d e s c r i b e s an evalua­ t i o n which i s being c a r r i e d out a t C o r n e l l on the a p p l i c a b i l i t y o f one such processor to l a r g e - s c a l e s c i e n t i f i c c a l c u l a t i o n s such as the simulations d e s c r i b e d e a r l i e r . The p a r t i c u l a r machine i s a F l o a t i n g Point Systems 190-L Array Processor (AP). The AP features i n c l u d e (6^7): (1) Independent storage f o r programs, data and constants. (2) Independent f l o a t i n g p o i n t m u l t i p l i e r and adder u n i t s . (3) Two blocks o f 32 f l o a t i n g p o i n t r e g i s t e r s . (4) Address indexing and counting by independent i n t e g e r a r i t h m e t i c u n i t with 16 i n t e g e r r e g i s t e r s . (5) P r e c i s i o n enhanced by 38 b i t i n t e r n a l f l o a t i n g p o i n t format, with hardware rounding. The very l a r g e i n s t r u c t i o n length and m u l t i p l e data paths permit p a r a l l e l o p e r a t i o n o f a l l o f these elements, so that f o r example a f l o a t i n g p o i n t add, f l o a t i n g point m u l t i p l y , branch on c o n d i t i o n , index increment, memory f e t c h data memory s t o r e , load to and from f l o a t i n g p o i n t and i n t e g e r r e g i s t e r s , can be simul­ taneously executed i n one 167 nanosecond c y c l e , although memory operations cannot be executed i n every c y c l e . The r e s u l t o f t h i s a r c h i t e c t u r e i s a processor with a maximum computation r a t e o f

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around 8 megaflops (6.* 7), or approximately the raw speed o f a CDC 7600 but i n the p r i c e range o f a minicomputer ($60,000 $140,000). Provided that such a processor can be made to operate near i t s p o t e n t i a l , t h i s c l e a r l y provides a very a t t r a c t i v e a l t e r n a t i v e to the maxi- or mini-computer s o l u t i o n s o u t l i n e d earlier. Since the AP cannot operate independently o f a host computer, the f i r s t d e c i s i o n that has to be made i s the appro­ p r i a t e type of host computer. E a r l i e r models of the AP (Model 120-B) have been attached e x c l u s i v e l y to minicomputers. The main concern with such a system i s whether the minicomputer i s s u f f i c i e n t l y powerful to keep the AP o p e r a t i n g near i t s f u l l p o t e n t i a l . The average AP c o n f i g u r a t i o n has only 32-64K o f data memory and 1-2K of program source memory. For l a r g e programs and large amounts o f data these l i m i t s imply a need to load s e c t i o n s (subroutines) o f the program i n t o the AP as necessary and to s i m i l a r l y segment and t r a n s f e r data to and from the AP. Unless the same t o t a l c a l c u l a t i o time than that r e q u i r e to using the AP except i f t h i s frees the m i n i f o r other t a s k s . In other words, assuming the m i n i i s a purchased machine, the key question i s s t i l l "How long w i l l i t take?". A number of i n s t a l l a t i o n s have been very s u c c e s s f u l l y o p e r a t i n g such a c o n f i g u r a t i o n i n a special-purpose environment f o r a few y e a r s . However, C o r n e l l appears to be the f i r s t i n s t a l l a t i o n to a t t a c h an AP to a l a r g e , general purpose host computer and attempt to make i t a v a i l a b l e to a wide range o f u s e r s . The C o r n e l l c o n f i g u r a t i o n c o n s i s t s o f an AP 190-L with 4K of Program Source Memory, 2.5K of Table Memory, and 96K of Main Data Memory, attached to an IBM 370/168 running the V i r t u a l Memory C o n t r o l Program. Jobs to be run i n the AP are prepared u s i n g the CMS i n t e r a c t i v e system and are then spooled to a s p e c i a l v i r t u a l machine c a l l e d the Array Processor Execution Manager (APEMAN) to which the AP i s normally attached. APEMAN schedules jobs which r e q u i r e the AP on the b a s i s of a scheduling a l g o r i t h m i n c l u d i n g requested p r i o r i t y and length of AP time (jobs can a l s o be placed i n hold and r e l e a s e d by the u s e r ) . When a job i s s e l e c t e d f o r i n i t i a t i o n , the designated user v i r t u a l machine i s autologged on, the AP i s l o g i c a l l y attached to that machine as a normal I/O device, and a designated CMS EXEC (command f i l e ) begins to execute i n the user's machine. At the end o f the run, or when the s p e c i f i e d time has elapsed, the AP i s detached from the user machine and returned to APEMAN, and the user machine i s logged o f f . The i n t e n t of the APEMAN scheduling, when complete­ l y implemented, i s the normal goal of any o p e r a t i n g system or c o n t r o l program; to have the resources of the AP as f u l l y occupied at a l l times as i s p r a c t i c a l . In a d d i t i o n , since the C o r n e l l Array Processor i s shared among s i x d i s t i n c t user groups, APEMAN i s used to monitor e q u i t a b l e d i s t r i b u t i o n of the AP f o r each aggregate group, and w i l l incorporate a p r i o r i t y scheme to facilitate this.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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The problem with a large c e n t r a l computer as host i s to r e s t r i c t as much as p o s s i b l e the time spent computing i n the host, not because o f the elapsed time i n v o l v e d , but because o f the cost differential. T h i s paper presents conclusions on the extent and d i f f i c u l t y o f reprogramming f o r the AP, the r e s u l t s o f benchmark comparisons f o r our Monte C a r l o c a l c u l a t i o n s , and on the p r e l i m i ­ nary assessments o f the host overhead, as w e l l as s t r a t e g i e s to minimize that overhead. Program Development The f i r s t c o n s i d e r a t i o n i s program development f o r the AP. For the AP to approach i t s p o t e n t i a l speed, the programmer has to attempt to keep as many o f the p a r a l l e l operations proceeding simultaneously as p o s s i b l e . A cross-assembler i s a v a i l a b l e to permit program development on the host computer o f AP machine code. The optimum s t r a t e g which i n v o l v e the bulk o r o u t i n e s executing i n the AP. While the subroutine i s executing i n the AP i t i s f e a s i b l e with c a u t i o n to have simultaneous execution o f code i n the h o s t . However, with a v i r t u a l memory environment l i k e VM, the r e s i d e n t host program w i l l be paged out of memory when i n a c t i v e and thus i n c u r very l i t t l e overhead, so we have made no attempt to provide synchronous o p e r a t i o n to date. A f t e r c r u c i a l s e c t i o n s o f the program have been optimized f o r the AP, outer loops can g r a d u a l l y be t r a n s f e r r e d from host code to AP code. The t r a d e o f f here i s that a large amount o f programmer time may be spent c o n v e r t i n g code which accounts f o r only 5-10% of the o v e r a l l execution time. In the case of our benchmarks, i n i t i a l l y f i v e loops ( l e v e l 0) were i d e n t i f i e d which are shown i n Table I . These are f a i r l y standard small loops which should be q u i t e s i m i l a r to those found i n most Monte-Carlo s i m u l a t i o n s . The inner loops were then incorporated i n an outer loop ( l e v e l 1) which generates one attempted move of each o f the p a r t i c l e s (a pass). For 200 p a r t i c l e s , the outer loop i s executed 200 times, and each inner loop i s a l s o executed 200 times f o r each c y c l e through the outer loop. E v e n t u a l l y , another outer loop ( l e v e l 2) w i l l be t r a n s f e r r e d to the AP which c a r r i e s out about 1000-5000 passes o f the p a r t i c l e s . Previous s t u d i e s have shown that the l e v e l 1 loop encompasses 99% o f the t o t a l execution time. Recoding the l e v e l 0 and l e v e l 1 loops, a t o t a l o f about 75 FORTRAN statements, i n v o l v e d about 100 hours by a programmer who began as a novice (but c l e a r l y f i n i s h e d as an e x p e r t ! ) . We estimate that t h i s time could be almost halved now. An important r e s t r i c t i o n o f the AP i s the l i m i t e d amount o f program source memory. The "expansion f a c t o r " observed f o r handcoding was o f the order o f 4-5 APAL statements f o r each FORTRAN statement. Since the maximum program source memory a v a i l a b l e f o r the AP i s 4K, t h i s gives a maximum FORTRAN subroutine length o f around 800 statements. I t i s a l s o c l e a r l y very d e s i r a b l e to not

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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TABLE I Inner Monte Carlo Loops Description DO 71 MM=1,NP Loop 71 c a l c u l a t e s the IF(MM.EQ.M)GOTO 71 energy change due to DX=SIDX2-DABS (DABS (XTRIAL-X (MM) ) -SIDX2)the t r i a l move o f a DY= SIDY2-DABS(DABS(YTRIAL-Y(MM))- SIDY2)particle and stores R2=DX*DX+DY*DY the new i n t e r p a r t i c l e LSAVE (MM)=R2 distances• LL= LSAVE (MM) C=R2 -DFLOAT (LL) C1=1.D0-C Loop 74 stores the E SAVE (MM)=C 1*RM ( LL )+C*RM ( LL+1) IND= INDEX (MINO (MM 71 DP^DPf ES (IND) -ESAV i n t e r p a r t i c l e distances f o r t r i a l moves which DO 74 MM=1,NP lower the energy o f IF(MM.EQ.M) GOTO 74 the system. GN (LSAVE (MM)=GN(LSAVE (MM) )+l.DO IND= INDEX (MINO (MM, M) ) +MAXO (MM, M) 74 ES(IND)=ESAVE(MM) DO 374 MM=1,NP IF(MM.EQ.M) GOTO 374 GN (LSAVE (MM) )=GN (LSAVE (MM) )+ACC DX= SIDX2 -DABS (DABS (X (M) -X (MM) ) - SIDX2 ) DY=SIDY2-DABS(DABS(Y(M) -Y(MM))-SIDY2) LLFDX*DX+DY*DY

GN(LL)=GN(LL)-ACCl IND= INDEX (MINO (MM, M) )+MAX0 (MM, M) 374 ES (IND=ESAVE (MM)

Loop 374 s t o r e s the new i n t e r p a r t i c l e energies and saves the i n t e r p a r t i c l e distances f o r t r i a l moves which r a i s e the energy o f the system.

DO 274 Mtt=l,NP IF(M.EQ.MM) GOTO 274 GN (LSAVE (MM)=GN (LSAVE (MM)+ACC DX=SIDX2-DABS (DABS (X(M) -X(MM) ) -SIDX2) DY=SIDY2-DABS(DABS(Y(M)-Y(MM))-SIDY2) LL=DX*DX+DY*DY 274 GN(LL)=GN(LL)+ACCl

Loop 274 stores the i n t e r p a r t i c l e distances f o r t r i a l moves which are r e j e c t e d .

DO 574 MM=1,NP IF(M.EQ.MM) GOTO 574 DX= SIDX2 -DABS (DAB S (X (M) -X (MM) ) - SIDX2 ) DY= SIDY2-DAB S(DABS(Y(M)-Y(MM))-SIDY2) LL=DX*DX+DY*DY 574 GN(LL)=GN(LL)+1.D0

Loop 574 stores the i n t e r p a r t i c l e distances f o r t r i a l moves which are r e j e c t e d w i t h an overwhelming probabi­ lity.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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ask a programmer to convert 750 FORTRAN statements which make up the r e s t o f the Monte-Carlo program used here, which only a f f e c t s 1% o f the execution time. A c c o r d i n g l y , C o r n e l l has developed an AP FORTRAN cross-compiler (APTRAN), running on the IBM 370/168. This compiler takes statements w r i t t e n i n a n a t u r a l "computations subset" o f FORTRAN, and produces APAL code or o p t i o n a l l y AP object code i n the a p p r o p r i a t e host format. The AP FORTRAN compiler i s only intended to remove the burden o f programming large amounts o f FORTRAN code which do not i n v o l v e the major p o r t i o n o f the computation time. Even though the compiler does have s e v e r a l l e v e l s o f o p t i m i z a t i o n , i t i s s t i l l envisaged that hand coding o f inner loops w i l l be b e n e f i c i a l . An added advantage o f the compiler i s that i t can reduce the amount o f program t r a n s f e r to and from the AP by c h a i n i n g c a l l s to AP subroutines. (See l a t e r ) However, because o f the complex­ i t y o f the compiler i t i s i n e v i t a b l y a q u i t e l a r g e program (more than 10,000 FORTRAN statements) f o r the compiler i s aroun memory i s an even more s e r i o u s l i m i t a t i o n f o r compiler-generated code • Results o f the Benchmarks Benchmarks on the AP were made by the use o f a simulator running on the 370/168. F l o a t i n g Point Systems provides a simulator f o r program development and debugging, as w e l l as f o r timing runs. Since the AP i s a completely asynchronous processor, such timings should be exact r e p r e s e n t a t i o n s o f the a c t u a l execu­ t i o n times. Simulators are n o t o r i o u s l y i n e f f i c i e n t , so that i t was necessary f o r long simulations to use a l o c a l l y modified v e r s i o n o f the F l o a t i n g Point Systems s i m u l a t o r . Because o f a d a p t a t i o n to the IBM 370 a r c h i t e c t u r e , t h i s simulator executes 3-4 times f a s t e r than the o r i g i n a l , and i n a l l runs to date i t has produced i d e n t i c a l r e s u l t s and timings. The longest run made f o r these benchmarks simulated a 22 m i l l i s e c o n d AP run, and consumed 60 seconds o f IBM 370/168 time. The r e s u l t s reported f o r other computers correspond to a c t u a l runs, and i n a l l cases the best o p t i m i z i n g FORTRAN compiler a v a i l a b l e was used, a t f u l l optimization l e v e l . Table I I shows the r e s u l t s obtained f o r some standard benchmarks (.5). The values f o r the CDC 7600 and H a r r i s / 4 are from the o r i g i n a l l i t e r a t u r e . In a l l cases, the timings are i n m i l l i ­ second, obtained from the time r e q u i r e d f o r 1000 i t e r a t i o n s o f the r e l e v a n t sequence o f i n s t r u c t i o n s . In the f l o a t i n g p o i n t benchmark, the AP performed only s l i g h t l y f a s t e r than the IBM 370/168. The f o l l o w i n g benchmark, i n v o l v i n g repeated c a l l s to e x t e r n a l a r i t h m e t i c f u n c t i o n s ( s p e c i f i c a l l y SQRT, ABS, SIN, ALOGIO, IFIX, ATAN, EXP) i s i n t e r e s t i n g since i t i l l u s t r a t e s the extremely low overhead i n c u r r e d by subroutine c a l l s i n the AP. The dot product benchmark shows the AP performing very f a v o r a b l y

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compared to the CDC 7600. This i s to be expected, since the p a r t i c u l a r s t r u c t u r e o f the a l g o r i t h m f o r a dot product i s i d e a l l y s u i t e d to the AP a r c h i t e c t u r e , which permits the simultaneous c a l c u l a t i o n o f the products, and t h e i r a d d i t i o n . In f a c t , i t i s p o s s i b l e t o w r i t e the e s s e n t i a l s t r u c t u r e o f a dot product loop i n one AP i n s t r u c t i o n . Table I I I shows the r e s u l t s achieved f o r the a c t u a l Monte Carlo inner loops ( l e v e l 0) d e t a i l e d i n Table I . The r e s u l t s f o r these inner loops show the AP 190L executing a t 0.6 times the speed o f the CDC 7600, or a t 2.5 times the speed o f the IBM 370/168. Table IV compares the AP 190L timings f o r the l e v e l 1 outer loop with those f o r the CDC 7600, IBM 370/168, and PRIME 400 computers. The execution time, T, o f the l e v e l 1 loop i s a n o n - l i n e a r f u n c t i o n o f the number o f p a r t i c l e s c o n s i d e r e d . I t i s given by the equation

where A and B have been determined f o r each machine. Timings which were obtained by e x t r a p o l a t i o n are i n d i c a t e d i n parentheses i n Table IV. This shows that as the number o f p a r t i c l e s i n c r e a s e s , the speed o f the AP 190L f o r the l e v e l 1 loop approaches a l i m i t of 0.63 times the speed o f the CDC 7600 o r 1.5 times the speed o f the IBM 370/168 and 67 times the speed o f the PRIME 400. We b e l i e v e the lower performance o f the AP 190L f o r the l e v e l 1 loop to be a t t r i b u t a b l e to a number o f f a c t o r s . Whereas i n s u i t a b l e a p p l i c a t i o n s the AP 190L has provided speeds above that o f a CDC 7600 (7), i n general the l i m i t i n g f a c t o r i n loop speed i s data memory access time. In t h i s i n i t i a l v e r s i o n o f the Monte C a r l o program, a great d e a l o f s t o r i n g and f e t c h i n g v a r i ­ ables and base addresses was performed between the l e v e l 0 loops. No attempt has yet been made to optimize the p o r t i o n s o f the l e v e l 1 loop around the inner loops. An a d d i t i o n a l c o n s i d e r a t i o n was the f a c t that the e x i s t i n g Monte Carlo FORTRAN program which was converted f o r the AP was not a t a l l o p t i m a l l y s t r u c t u r e d f o r the AP. I t i s the case that the r e s u l t s f o r the AP were obtained from hand coded r o u t i n e s , while those f o r the CDC 7600 and IBM 370/168 computers were obtained from the FTN4 and H Extended o p t i m i z i n g compilers r e s p e c t i v e l y . However i n our experience these o p t i m i z i n g compilers have proved extremely e f f i c i e n t f o r the small loops i n v o l v e d here, while there are extensive opportu­ n i t i e s f o r f u r t h e r hand o p t i m i z a t i o n o f the AP code. For these reasons we b e l i e v e the f a c t o r o f 0.42 times the CDC 7600 speed to be a lower l i m i t o f the c a p a b i l i t y o f the AP f o r t h i s a p p l i c a ­ tion.

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TABLE I I General Benchmarks CDC 7600 Integer A r i t h .

1.8

F l o a t i n g Point

0.21

E x t e r n a l Funct.

0.32

Dot

0.70

Product

Harris/4

IBM 370/168

41

AP 190L

4.3

NA

4.8

0.60

0.55

4.1

0.97

0.26

1.50

0.84

30

These benchmarks are taken from r e f e r e n c e 5.

TABLE I I I Monte C a r l o Benchmarks LOOP

CDC 7600 TIME (MS)

IBM/ 370 TIME (MS)

AP TIME (MS)

71

.597

2.2

74

.316

1.2

374

.583

1.9

574

.487

1.4

274

.448

1.6

.885 .634 .885 .300 .601 TOTAL

2.431

8.3

3.305

A l l times are i n m i l l i s e c o n d s and measure the time needed to execute the s t a t e d loop 100 times.

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TABLE IV Comparativ N - P a r t i c l e Monte Carlo Program (milliseconds) //Particles

PRIME 400

IBM 370/168

CDC 7600

AP 190L

16

(416.0)

6.8

(2.0)

4.9

36

1650.0

30.2

(9.2)

21.9

64

4710.0

98.2

28.0

(66.3)

67.2

(158.5)

100 1000 10000

3

(1.09 1 0 ) 6

(1.00 10 ) 7

(9.92 1 0 )

243.0 4

(6.55 1 0 )

6

(6.53 1 0 )

(2.32 1 0 ) (2.31 1 0 )

3

5

4

(1.53 1 0 ) 6

(1.52 1 0 )

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Host Overhead The f u l l cost-recovery r a t e f o r the AP-190L at C o r n e l l , based on a three-year a m o r t i z a t i o n schedule, i s $40/hour. Since the previous benchmarks have shown that the AP can run at from 1.5-2.5 times the speed of the IBM 370/168, t h i s y i e l d s a c o s t comparison of about 80:1 i n the C o r n e l l environment. It i s impor­ tant to note that t h i s r a t i o uses the prime-time 370 r a t e s , and can be reduced by a f a c t o r of about 0.4 f o r overnight turnaround. The comparison may a l s o be s u b s t a n t i a l l y l e s s favorable to the AP f o r a p a r t i a l l y - s u b s i d i z e d host computer. However, t h i s i s s t i l l a dramatic demonstration of the p o t e n t i a l cost advantage of c a r r y i n g out l a r g e - s c a l e s c i e n t i f i c c a l c u l a t i o n s on such an attached p r o c e s s o r . The magnitude o f the c o s t - d i f f e r e n t i a l makes the problem o f host overhead even more c r u c i a l , since the cost of a run with 907 of the computation i n th host could s t i l l be t o t a l l overhead i s i n c u r r e d i n a number o f ways: (i) Job submission. (ii) System overhead i n making the AP a v a i l a b l e to the p a r t i c u l a r user. (iii) Maintenance of a main program r e s i d e n t i n the host (AP programs can only be executed as subroutines c a l l e d from the h o s t ) . (iv) CALL to the AP subroutine from the main program. (v) I n i t i a l i z a t i o n o f the AP. (vi) T r a n s f e r of program source microcode to the AP f o r every subroutine c a l l , i n c l u d i n g the updating o f l o c a t i o n tables, etc. (vii) T r a n s f e r of data to and from the AP. (viii) Release the AP f o r a subsequent CALL. (ix) RETURN from the AP subroutine. (x) Release the AP f o r other u s e r s . Table V d e t a i l s the t y p i c a l host overhead i n v o l v e d i n the execution of an AP program a t C o r n e l l . Due to the c h a r a c t e r i s t i c s of the C o r n e l l V i r t u a l Machine environment mentioned e a r l i e r , there i s no a p p r e c i a b l e overhead ( l e s s than 1 msec) to maintain a main program r e s i d e n t i n the host, since the program i s paged out of r e a l memory when i n a c t i v e . I t i s important to note that the m a j o r i t y o f the c o n t r i b u t i o n s to the t o t a l overhead are f i x e d q u a n t i t i e s . Exceptions which are under the c o n t r o l o f the pro­ grammer i n c l u d e the number of data t r a n s f e r s (APPUTs and APGETs), and the number o f subroutine c a l l s . One major o b j e c t i v e o f the C o r n e l l APTRAN compiler and i t s a s s o c i a t e d linkage e d i t o r was to minimize data t r a n s f e r s by t r a n s f e r r i n g data i n COMMON blocks i n one chained o p e r a t i o n (without e x p l i c i t t r a n s f e r being r e q u i r e d by the programmer, as i s otherwise the c a s e ) . In a d d i t i o n , APTRAN chains AP subroutine c a l l s together, e f f e c t i v e l y r e p l a c i n g the CALL and RETURN, program t r a n s f e r , and data t r a n s f e r ( i n some o

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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TABLE V T y p i c a l Values f o r Host Overhead

(IBM 370/168)

Operation

Approx. Overhead (msec.)

Comment s

APM

10

Job submission

SUBMIT

AP ATTACH

Subroutine CALL

0.5

APINIT

58

I n i t i a l i z e the AP

APPUT .

1

Data T r a n s f e r to AP (each)

APGET

Data T r a n s f e r from AP (each)

T r a n s f e r AP code

10

Put r o u t i n e ' s micro­ code i n AP (up to 4K instructions) Release AP f o r subsequent CALL

APRLSE

Subroutine RETURN

0.5

APDETACH

10

TOTAL

132 msec.

R e l i n q u i s h AP from user machine

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MATTER

cases) by an AP "JUMP SUBROUTINE" i n s t r u c t i o n . This i n s t r u c t i o n was shown e a r l i e r to i n c u r an extremely low overhead (167 nano­ second worst c a s e ) . The Table shows that a minimum overhead of about 150 msec, o f 370/168 cpu time must be i n v o l v e d i n any AP computation, no matter how t r i v i a l . The cost r a t i o of 80:1 leads to a r u l e - o f thumb estimate that the f u l l cost b e n e f i t s of the AP w i l l not be approached unless a subroutine can be created which: ( i ) Contains l e s s than 300-400 FORTRAN statements (because o f the l i m i t e d Program Source Memory) i f u s i n g a compiler. T h i s number might be doubled f o r hand-coding. (ii) Operates on l e s s than 64K f l o a t i n g p o i n t numbers as data at one time (96K f o r hand-coding). (iii) Computes f o r at l e a s t 1-2 minutes i n the AP (3-5 minutes on an IBM 370/168 or 30-90 seconds on a CDC 6700) between subroutine CALL and RETURN. (This does not imply that intermediate dat the AP r e s i d e n t s u b r o u t i n e ) Point ( i i i ) above may need some c l a r i f i c a t i o n ; the l a r g e s t part o f the overhead per subroutine c a l l i s i n v o l v e d i n i n i t i a l i ­ z i n g the AP and t r a n s f e r r i n g the microcode f o r the subroutine. Once the subroutine i s o p e r a t i n g i n the AP, i t can make data t r a n s f e r s to and from the host at an overhead of about 1 msec, per t r a n s f e r (independent o f the amount o f data t r a n s f e r r e d up to the c a p a c i t y of the AP). Thus c a r e f u l programming can ensure that the same subroutine remains r e s i d e n t i n the AP, but that t h i s subroutine operates on sucessive batches of data; i t i s the r e t u r n from the subroutine to the host and the corresponding c a l l which must be minimized. For a p p l i c a t i o n s which f i t the above c o n s t r a i n t s the cost advantages of the AP can be outstanding. For example, i n the Monte-Carlo c a l c u l a t i o n s described i n t h i s paper the o v e r a l l program s t r a t e g y i s s t i l l being r e f i n e d to minimize subroutine c a l l s , but i t appears that a goal of over 997o of the computation time i n the AP and l e s s than 17o 370/168 overhead can d e f i n i t e l y be r e a l i z e d . This i n turn implies an o v e r a l l cost i n the range o f $50/AP hour, even using the prime time r a t e s on the IBM 370/168. Accuracy o f the

Calculations

One concern w i t h p o t e n t i a l users of the AP 190L (or the equi­ v a l e n t AP 120B) f o r s c i e n t i f i c work i s the a v a i l a b l e word l e n g t h . No hardware double p r e c i s i o n c a p a b i l i t y i s a v a i l a b l e on the AP 190L, so that a l l f l o a t i n g p o i n t operations are c a r r i e d out i n the 38 b i t i n t e r n a l f l o a t i n g p o i n t format, with a 10 b i t b i n a r y exponent, a 28-bit mantissa, and a hardware convergent rounding algorithm i n the F l o a t i n g Adder and F l o a t i n g M u l t i p l i e r . Floating Point Systems (6) s t a t e that t h i s provides a p r e c i s i o n of 8.1 decimal d i g i t s , as compared to 7.2 decimal d i g i t s f o r IBM 370 s i n g l e p r e c i s i o n , 16.8 decimal d i g i t s f o r IBM 370 double p r e c i s i o n ,

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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and approximately 14 decimal d i g i t s f o r the CDC 7600 i n s i n g l e precision. Table VI shows the r e s u l t s o f some accuracy comparisons which we have made. The dot product r e s u l t s correspond t o a dot product o f two v e c t o r s o f 1000 elements, with random values between 0 and 1. This shows that the AP 190L r e s u l t gives the f i r s t 7 f i g u r e s o f the r e s u l t s from the IBM 370 double p r e c i s i o n calculation. The other e n t r i e s i n Table VI show r e s u l t s from the Monte Carlo c a l c u l a t i o n s . TABLE VI E f f e c t o f AP Word Length on P r e c i s i o n IBM REAL*8 Dot

Product

AP 190L

0.255178129E0

T o t a l Energy (16)

-30.7006546

-30.7006483

T o t a l Energy (36)

-61.6051765

-61.6052179

The t o t a l energy i s the running average energy computed f o r one pass through the system o f 16 or 36 p a r t i c l e s . M a i n t a i n i n g the accuracy o f the running average energy i s c r u c i a l i n Monte Carlo calculations. For l a r g e systems o f s e v e r a l hundred p a r t i c l e s i t may be necessary to recompute the running average i n the course o f a pass, since t h i s q u a n t i t y i s the most s e n s i t i v e to accumu­ l a t e d e r r o r s . This shows that the AP 190L r e s u l t gives the f i r s t 7 f i g u r e s o f the IBM 370 double p r e c i s i o n r e s u l t f o r 16 p a r t i c l e s , and the f i r s t 6 f i g u r e s f o r 36 p a r t i c l e s . A f t e r t e s t i n g the AP random-number generator r o u t i n e a v a i l ­ able i n the F l o a t i n g Point Systems Math L i b r a r y , a random-number generator f o r a 28-bit mantissa a r c h i t e c t u r e was w r i t t e n i n FORTRAN and compiled on the APTRAN compiler. Conclusions In our experiments the F l o a t i n g Point Systems 190L Array Processor proved capable o f speeds between 0.43 and 0.66 the speed o f a CDC 7600. The broad c o n c l u s i o n i s that the AP 190L appears to be between 10 and 100 times more c o s t - e f f e c t i v e than the other systems we have discussed f o r the type o f Monte Carlo simulations we r e p o r t . In f a c t the c o s t - e f f e c t i v e n e s s o f the AP 190L i s such that there i s l i t t l e economic i n c e n t i v e to extensive program o p t i m i z a t i o n , though i t may be e s t h e t i c a l l y s a t i s f y i n g . While i t w i l l be necessary l a t e r to i n c l u d e the a d d i t i o n a l costs i n v o l v e d when a c t u a l l y u s i n g the combined IBM 370/AP 190L system, we do not estimate that these costs w i l l m a t e r i a l l y a l t e r our c o n c l u s i o n . I t appears a l s o that the word

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length o f the AP 190L i s s u f f i c i e n t to provide adequate f o r our purposes.

precision

Acknowledgement s This work was supported by the N a t i o n a l Science Foundation Grant No. DMR-74-23494 and a l s o through the M a t e r i a l s Science Center, C o r n e l l U n i v e r s i t y , through the N a t i o n a l Science Founda­ t i o n Grant No. DMR-72-03029. The O f f i c e o f Computer Services provided IBM 370/168 computer time. We would a l s o l i k e to thank Mr. J . Tobochnik f o r help i n running the benchmarks on the PRIME computer. Literature 1. 2. 3. 4. 5. 6. 7.

Cited

Ceperley, D., Chester, G.V. and Kalos, M.H., Phys. Rev., (1977), 16, 3081. Ceperley, D., Chester (1978), 17, 1070. Gann, R., Chakravarty, S. and Chester, G.V., Phys. Rev., (1978), to be p u b l i s h e d . M e t r o p o l i s , N., Rosenbluth, A.W., Rosenbluth, M.N., T e l l e r , A.H. and T e l l e r , E., J. Chem. Phys. (1953), 21, 1087. Schaefer, H.F. and Miller, W.H., Computers and Chemistry, (1971), 1, 85. F l o a t i n g Point Systems "AP 120-B Processor Handbook", 7259-02. F l o a t i n g Point Systems, Beaverton, Oregon 1976. Bucy, R.S. and Senne, K.D., "Nonlinear Filtering Algorithms f o r P a r a l l e l and Pipe Line Machines". Proceedings o f the Meeting on P a r a l l e l Mathematics - P a r a l l e l Computers, Munich March 1976, North Holland Press, Amsterdam.

RECEIVED August 15,

1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

11

Simulating the Dynamic and Equilibrium Properties of a Multichain Polymer System DAVID E. KRANBUEHL and BRUCE SCHARDT Department of Chemistry, College of William and Mary, Williamsburg, VA 23185

The dynamic behavio subject of numerous molecula ature. The b a s i s f o r most of t h i s work is the bead-and-spring model of Rouse (1). In the Rouse model, a polymer chain i s rep­ resented by a l a r g e number o f connected segments. Each segment is statistically equivalent t o a sufficient number of monomer u n i t s such t h a t the o v e r a l l motion of the polymer may be ideal­ ized by the movement o f these f r e e l y j o i n t e d l i n k s . T h i s model with its a s s o c i a t e d m o d i f i c a t i o n s f o r hydrodynamic i n t e r a c t i o n and excluded volume has achieved considerable success i n d e s c r i b ­ ing dielectric, neutron s c a t t e r i n g , nuclear magnetic and v i s c o e l a s t i c - m e c h a n i c a l p r o p e r t i e s of polymer systems (2-10). A major l i m i t a t i o n o f these a n a l y t i c a l c a l c u l a t i o n s is the difficulty of i n t r o d u c i n g s p e c i f i c molecular i n t e r a c t i o n s without approximating or pre-averaging parameters of p h y s i c a l i n t e r e s t . Another approach used t o i n v e s t i g a t e dynamic behavior i s t o simulate Brownian motion u s i n g Monte Carlo techniques on a lat­ tice-model polymer chain with the a i d of a high speed digital computer (11, 12, 13, 14, 15). In our lattice model the c o n f i g ­ u r a t i o n of a polymer chain N - l u n i t s long is represented by a s t r i n g of N connected p o i n t s on a 3-dimensional lattice. The p o i n t s r e f e r r e d to as beads lie on the v e r t i c e s of the lattice. The l i n k s between the beads r e p r e s e n t i n g a l a r g e number of monomer u n i t s are each of u n i t l e n g t h . Brownian motion of the chain r e ­ s u l t i n g from random collisions with solvent molecules i s simulated by choosing beads at random and then moving them t o a new p o s i t i o n on the lattice according to f i x e d bead movement r u l e s which main­ t a i n chain c o n n e c t i v i t y and a bead segment d i s t a n c e of one lat­ tice unit. As this process, which i s c a l l e d a bead c y c l e , is re­ peated again and again, the chain moves from one conformation t o another, e v e n t u a l l y ( i n p r i n c i p l e ) t a k i n g on all the conformations i n its e q u i l i b r i u m ensemble. P e r i o d i c sampling o f the p o s i t i o n s of the beads and c a l c u l a t i o n of q u a n t i t i e s o f experimental i n t e r ­ est give estimates o f e q u i l i b r i u m ensemble averages. Relaxation behavior i n the e q u i l i b r i u m ensemble i s s t u d i e d by s t o r i n g these

0-8412-0463-2/78/47-086-125$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

126

COMPUTER MODELING OF MATTER

q u a n t i t i e s along with the elapsed time i n bead c y c l e s and then c a l c u l a t i n g t i m e - c o r r e l a t i o n functions f o r the chain p r o p e r t i e s of i n t e r e s t . Bead c y c l e time may be r e l a t e d t o r e a l time by measur­ ing the t r a n s l a t i o n a l d i f f u s i o n constant of the polymer chain. We regard these dynamic Monte Carlo l a t t i c e chain simulations as computer "experiments" which complement a n a l y t i c a l c a l c u l a ­ t i o n s . A l a t t i c e chain model has been widely used i n both a n a l y t ­ i c a l and numerical c a l c u l a t i o n s to d e s c r i b e e q u i l i b r i u m p r o p e r t i e s of polymers (1(5). Numerical c a l c u l a t i o n s have been p a r t i c u l a r l y h e l p f u l i n t r e a t i n g e q u i l i b r i u m p r o p e r t i e s which are d i f f i c u l t to handle a n a l y t i c a l l y (l£, 18^ 1 £ ) . S i m i l a r l y , the Monte Carlo l a t t i c e chain model i s o f t e n more advantageous than the a n a l y t ­ i c a l model of Rouse i n the treatment of dynamic behavior. The dynamic behavior of the Monte Carlo l a t t i c e model i s s i m i l a r t o the Rouse model (20, 21, 22). However, the Monte C a r l o l a t t i c e model f a c i l i t a t e s the i n t r o d u c t i o n of s p e c i f i c types of i n t e r ­ a c t i o n such as excluded which are not r e a d i l y amenabl Thus, the Monte C a r l o l a t t i c e chain model should provide a u s e f u l approach f o r studying the dynamic and e q u i l i b r i u m p r o p e r t i e s of polymers as a f u n c t i o n of c o n c e n t r a t i o n , c u r r e n t l y an important and p a r t i c u l a r l y d i f f i c u l t problem i n polymer chemistry. Model D e t a i l s of the dynamical model f o r a s i n g l e l i n e a r polymer chain on a simple cubic l a t t i c e have been p r e v i o u s l y described by V e r d i e r (2%). The model has been extended so that the dynamic p r o p e r t i e s of m u l t i - c h a i n polymer systems can be simulated. In the model, a polymer chain i s represented by a s t r i n g of N con­ nected p o i n t s on a cubic l a t t i c e . A c o l l e c t i o n of n polymer chains each of l e n g t h N are placed i n a h y p o t h e t i c a l cubic space of volume L3 with penetrable w a l l s . When a polymer bead moves through one of the w a l l s , i t i s r e l o c a t e d a distance L so that i t reappears on the opposite s i d e of the box. Thus, the number of beads i n the volume remains constant. This procedure i n t r o ­ duces an a r t i f i c i a l p e r i o d i c i t y of p e r i o d L i n t o the system as shown i n Figure 1. Brownian motion of the chains i s simulated by choosing beads i n the box at random. For non-end beads, the chosen beads p o s i ­ t i o n r j i s moved to a new p o s i t i o n r j according to r

j

=

r

j-l

+

r

r

j + l - J>

where r ^ i s a v e c t o r from an a r b i t r a r y p o s i t i o n to the j t h bead. I f an end bead i s p i c k e d , i t moves to one of four p o s s i b l e new p o s i t i o n s r^ ^ such that n

(^end

- r

n e x

t)

#

(*end - r

n e x

t ) = 0.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

11. KRANBUEHL AND SCHARDT

p-o-6

j

Properties of a Multichain Polymer System 127

1

O-O-J

^

s I

o-o-A

1

o-o-J

p-o-A

p-o-J

i

o-o-o

pK>-0

o-o-<J

1

5

Figure 1. Two eight-bead chains

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

128

This process, which may be thought of as corresponding t o 1/nN u n i t s of time, i s r e f e r r e d t o as a bead c y c l e . The e f f e c t o f c h a i n entanglements and excluded volume i s accounted f o r by not a l l o w i n g the bead t o move t o i t s new s i t e i f t h i s s i t e i s already occupied by another bead from one of the chains. In t h i s study r e l a x a t i o n from e q u i l i b r i u m p o s i t i o n s was exam­ i n e d . F i v e t o t e n computer runs each were made on m u l t i - c h a i n systems f o r N=10 and N=20 at v a r y i n g c o n c e n t r a t i o n s . The f i n a l c o n f i g u r a t i o n o f the chains i n each run served as the i n i t i a l con­ f i g u r a t i o n f o r the next run. The t o t a l time i n t e r v a l of each run c o n s i s t e d of 100 t o 1000 equal s u b i n t e r v a l s . Each s u b i n t e r v a l or frame l a s t e d between 1.0 to 1.5 nN^ bead c y c l e s , a time which i s comparable t o the r e l a x a t i o n f o r the v e c t o r end-to-end l e n g t h . Since v e c t o r end-to-end l e n g t h appears t o r e l a x on the time s c a l e of the slowest motions of a c h a i n , the number of frames represents a crude estimate of the number of independent samples represented i n each run. At the s t a r t o f eac taneous value f(t ) of the v e c t o r end-to-end f o r each chain i n the box and i t s square Jf^(t ) were sampled and saved. At each of a s e r i e s of l a t e r time i n t e r v a l s t + t , / and j ^ were again sampled and m u l t i p l i e d by the corresponding values at t t o g i v e sampled values of the products / ( t ) • I ( t + t ) f o r each c h a i n . These products along with sampled values of ^ obtained at the same time were added i n t o running sums. At the end of each run, the running sums were converted t o averages and used to give estimates of the ensemble averages and . The estimates from a l l runs and a l l the chains i n the box were combined. The a u t o c o r r e l a t i o n f u n c t i o n i n / was c a l c u l a t e d as G

Q

2

Q

Q

Q

Q

2

2

Q

G

/>(/,/,t) =/.

Results and D i s c u s s i o n 2

2

Table I gives the values of / , * < and f o r each simula­ tion. The estimate of the u n c e r t a i n t y i s the standard d e v i a t i o n of the average over a l l the runs. The values of the expansion f a c t o r squared are computed a s * = < | V ( N - l ) , where N i s the number o f beads per c h a i n . The bead d e n s i t y or polymer volume f r a c t i o n V i s the t o t a l number of beads d i v i d e d by the number of lattice sites. A l l values of < / > f o r the m u l t i p l e 10 and 20 bead s i m u l a t i o n s agree t o w i t h i n one standard d e v i a t i o n of p r e ­ v i o u s l y p u b l i s h e d e q u i l i b r i u m data. De Vos and Bellemans (26) examined the c h a i n l e n g t h depend­ ence of<£ >for N=T, 11, 21 and 31 bead chains as a f u n c t i o n of the polymer volume f r a c t i o n ^ . More r e c e n t l y , W a l l , Chin and Mandel (27) have examined < £ > f o r chains up t o 105 beads. The dependence of
2

2

2

2

2

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

KRANBUEHL AND SCHARDT

Properties of a Multichain Polymer System 129

CO hO\H O V

t«-

t-ao vo

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o o o o on on

t—_3-

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LA LA LA

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

130

COMPUTER MODELING OF MATTER

Time

Figure 2. ln( ) vs. time in units of nN bead cycles, (a) Multiple 10-bead chains. Polymer volume fraction . (U) 0.0195, (%) 0.1852, (\J) 0.4167, (O) 0.6122. (b) Multiple 20-bead chains. Polymer volume fraction (p. (U) 0.0391, (%) 0.1953, O 0.4082,(0)0.5248. 3

P

v

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

11.

KRANBUEHL AND SCHARDT

Properties of a Multichain Polymer System 131

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER

132

MODELING OF MATTER

2

l i m i t 9=1. Bellemans' data suggests that < £ > ~ (N-l)l-OT when 9=1. This should be compared with <^2>~(N-1 ) for a s i n g l e chain with excluded volume and < ( > ~ N - 1 f o r a Gaussian chain. A n a l y t i c a l and experimental data support the c o n c l u s i o n that the e q u i l i b r i u m value o f < ( > approaches gaussian behavior i n the bulk phase (28_, 2 £ ) . F l o r y f i r s t proposed t h i s r e s u l t and argued that i n the bulk phase the polymer chain has nothing t o gain by expanding s i n c e a decrease i n i n t e r a c t i o n with i t s e l f i s compensated f o r by increased i n t e r f e r e n c e with i t s neighbors. The a u t o c o r r e l a t i o n f u n c t i o n s / , t ) of v e c t o r end-to-end l e n g t h f o r 10 and 20 bead chain systems at v a r i o u s bead d e n s i t i e s are shown i n Figure 2. The p l o t shows l n / 0 ( / , / , t ) versus time i n nN^ bead c y c l e s . N i s the number of beads i n the chain and n i s the number o f chains on the l a t t i c e . Although the a u t o c o r r e l a ­ t i o n f u n c t i o n s ^ ( / , / , t ) as shown i n F i g u r e 2 are not simple expo­ n e n t i a l s , we a r b i t r a r i l y d e f i n e a r e l a x a t i o n time 7^_/ as the time i n t e r v a l i n w h i c h l/e. In order t o c h a r a c t e r i z end-to-end l e n g t h , we have e x t r a c t e d two other time parameters from each a u t o c o r r e l a t i o n f u n c t i o n i n F i g u r e 2 . At long times, the semilog p l o t s appear t o approach l i n e a r i t y . We estimate the longest r e l a x a t i o n time T and i t s c o n t r i b u t i o n A t o the r e l a x a ­ t i o n from the l i m i t i n g slope o f l n / 0 ( / , / , t ) versus time. The l i m i t i n g slope r e l a x a t i o n time'Tg should be dependent on the longer range cooperative motion o f the chains. Thus, 7T i s the parameter o f g r e a t e s t i n t e r e s t i n examining the e f f e c t o f concen­ t r a t i o n on the dynamic behavior o f the polymer c h a i n . 1

#

2

0

2

2

e

/^T^,/,^/) =

s

S

Table I I l i s t s the values of A and 7^ and 7^/ f o r the 10 and 20 bead m u l t i p l e chain simulations i n u n i t s o f nN^ bead cy­ cles. For chains without excluded volume,"?^ i n u n i t s o f nN bead c y c l e s i s approximately constant. As shown i n Table I I , we observe f i r s t t h a t T ^ increases as the polymer volume f r a c t i o n i n c r e a s e s . For the 20 bead c h a i n s , the c o n t r i b u t i o n o f the l o n g ­ est r e l a x a t i o n t i m e t o /,t) as measured by A increases back toward the no excluded volume r e s u l t as the polymer volume f r a c t i o n i n c r e a s e s . No t r e n d i n A f o r the 10 bead chains i s observed. F i g u r e 3 shows a p l o t o f 7^ versus the polymer volume f r a c ­ t i o n *p. The n o n - l i n e a r behavior of *2" versus 9 r e f l e c t s the non­ uniform d e n s i t y of polymer beads on the l a t t i c e at low concentra­ tions. At low values o f 1P the beads are c l u s t e r e d i n n u n i t s , the number o f chains, each o f N beads. T h i s makes the l o c a l bead d e n s i t y much greater than the l a t t i c e average. As the polymer volume f r a c t i o n 9 i n c r e a s e s , the l o c a l bead d e n s i t y does not r i s e as f a s t . As 9 i n c r e a s e s f u r t h e r , a point i s reached when i n ­ c r e a s i n g the bead d e n s i t y begins t o have a pronounced e f f e c t on the number o f i n t e r c h a i n excluded volume c o n f l i c t s . At t h i s p o i n t , *2*s r i s e s sharply with an i n c r e a s e i n 9. Free volume has been a p a r t i c u l a r l y u s e f u l concept to de­ s c r i b e dynamic p r o p e r t i e s o f polymer systems. Assuming that the e

3

s

9

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

KRANBUEHL AXD SCHARDT

Properties of a Multichain Polymer System

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LA

I I

I

VO

O O O O O

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O

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Eh O

OJ

OJ OJ

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O

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00

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LA

I o Eh

I

f2 o Eh O W >

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< •H

>>

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^cu cu CJ o o cs C * -p •H

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o w Eh

LA ON L A C"-— r l CO r l O H O H 4 V O V D

ON L A 0 O L A OO ON O OJ

o ooo o

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H -=r ON oo H

H

0 0 0 0 0

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In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

133

134

COMPUTER MODELING OF MATTER

0.0

0.5

1.5

1.0

1 - *>

Figure 3.

ln(r ) VS. c /(l -
f

= 10 chains, (U) N = 20 chains.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

11.

Properties of a Multichain Polymer System 135

KRANBUEHL AND SCHARDT

dependence o f ^ and the v i s c o s i t y are s i m i l a r l y dependent on long time c o o p e r a t i v e motions o f the polymer c h a i n s , we may w r i t e as suggested by D o o l i t t l e (30) and W i l l i a m s , Landel and F e r r y (8_, 51) T

s

= A exp ( B V / V ) = A exp ( B ^ ( l - ^ ) Q

f

where A and B are constants, V i s the volume occupied by the polymer molecule, and V f i s the average f r e e volume per molecule. Table I I l i s t s values o f I n (7^) a n d p / ( l - f ) . F i g u r e 3 shows t h a t a p l o t o f l n ( t ^ ) versus f * / ( l - W f o r the 10 and 20 bead m u l t i p l e c h a i n systems i s l i n e a r as p r e d i c t e d by the theory. Thus, over the range o f polymer volume f r a c t i o n s s t u d i e d , the f r e e volume theory provides a good d e s c r i p t i o n o f the dependence o f the r e l a x ­ a t i o n time on polymer volume f r a c t i o n . Only two chain lengths have been examined. I t i s appropriate to make only t e n t a t i v e observations about the N dependence o f as the d e n s i t y f# o f th s i n g l e c h a i n systems wit 7£*Nr. T h i s r e s u l t may be compared with the Rouse, no excluded volume r e s u l t f o r a s i n g l e c h a i n t h a t ^ ^ N . Comparing the 1 0 and 20 bead c h a i n systems with excluded volume, the N dependence of does not appear t o be a f f e c t e d by an i n c r e a s e i n 0 i n the c o n c e n t r a t i o n range s t u d i e d . Further work a t higher concentra­ t i o n s and f o r longer chain lengths N i s i n progress. Q

3

P a r t i a l f i n a n c i a l support from the Petroleum Research Corpo­ r a t i o n administered by the American Chemical S o c i e t y i s g r a t e ­ f u l l y acknowledged. Literature Cited 1. 2. 3. 4. 5. 6.

Rouse, P. E., Jr., J. Chem. Phys. (1953) 2 1 , 1272. Zimm, B. H., Jr., J. Chem. Phys. (1956) 24, 269. Pyun, C. W. and Fixman, M., J. Chem. Phys. (1965) 42, 3838. Tshoegl, N. W., J . Chem. Phys. (1964) 40, 473. De Gennes, P. G., Macromolecules (1977) 9, 587 and 594. Stockmayer, W. H., " F l u i d e s M o l e c u l a i r e s " (Gordon Breach, New York, 1 9 7 6 ) . 7. Gupta, S., Shah, V. and Forsman, W., Macromolecules (1974) 7,

948. 8. 9.

F e r r y , J. D., " V i s c o e l a s t i c P r o p e r t i e s o f Polymers" (Wiley, New York, 1 9 7 0 ) . McCrum, N. G., Read, B. F. and W i l l i a m s , G., " A n e l a s t i c and D i e l e c t r i c E f f e c t s in Polymeric S o l i d s " (Wiley, New York, 1967).

10. 11.

Stockmayer, W. H., Pure Appl. Chem. (1967) 15, 539. V e r d i e r , P. H. and Stockmayer, W. H., J. Chem. Phys. (1962) 36,

227.

12. V e r d i e r , P. H., J. Chem. Phys. (1966) 45, 2122. 13. V a l e u r , B., J a r r y , J., Geny, F. and Monnerie, L., J. Polym.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

136

COMPUTER MODELING OF MATTER

Sci., Poly. Phys. (1975) 13, 667 and 675. 14. Taran, Yu A., Dokl. AN SSR (1970) 191, 1330. 15. Kranbuehl, D. E. and V e r d i e r ,P.H.,J.Chem. Phys.

3145.

(1972) 56,

16. Yamakawa, H., "Modern Theory o f Polymer S o l u t i o n s " (Harper &

Row, New York, 1971).

17.

W a l l , F., Chin, J . and Mandel, F., Macromolecules

3143.

(1977) 7,

18. Rubin, R. and Weiss, G., Macromolecules (1977) 10, 332. 19. Gobush, W., S o l c , K. and Stockmayer, W.H.,J.Chem. Phys.

(1974)60, 12. 20. 21. 22.

O r w o l l , R. A. and Stockmayer, W. H., Advan. Chem. Phys.

(1969)

15, 305.

V e r d i e r ,P.H.,J.Chem. Phys. (1970) 52, 5512. Iwata, K. and Kurata,M.,J.Chem. Phys. (1969) 23. Kranbuehl, D., V e r d i e r

50, 4008.

(1973)59,3861.

24. Kranbuehl, D. and V e r d i e r ,P.,J.Chem. Phys. (1977) 67, 36l. 25. V e r d i e r ,P.,J.Comp. Phys. (1969)4,204. 26. De Vos, E. and Bellemans, A., Macromolecules (1975) 8, 651. 27. W a l l , F., Chin, J. and Mandel,F.,J.Chem. Phys. (1977) 66,

28.

3143.

Fixman, M. and Peterson,J.,J.Am. Chem. Soc.

3524.

(1964) 86,

7,

29. C o l t o n ,J.etal,Macromolecules (1974) 863. Doolittle, A., J. Appl. Phys. (1951) 22, 1471; ibid. (1952) 23, 236. 31. W i l l i a m s , M., Landel, R. and F e r r y ,J.,J.Am. Chem. Soc. (1955) 77, 3701.

RECEIVED August 15, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

12

Application of Conformational Energy Calculations to Defect Properties in Polymer Crystals RICHARD H. BOYD Department of Materials Science and Engineering, and Department of Chemical Engineering, University of Utah, Salt Lake City, UT 84112

The conformationa has proven to be h i g h l stabilities and other p r o p e r t i e s o f i s o l a t e d organic molecules (1). The model b a s i c a l l y assumes t h a t molecular e n e r g e t i c s (conformational energy) can be represented by e m p i r i c a l t r a n s f e r ­ able energy f u n c t i o n s that d e s c r i b e bond s t r e t c h i n g , bending and t w i s t i n g and a l s o non-bonded i n t e r a c t i o n s . Molecular geometries are those o f local minima in the t o t a l conformational energy. Thus the problem of s t r u c t u r e c a l c u l a t i o n becomes that o f seeking l o c a l minima i n a f u n c t i o n with v a r i a b l e s equal t o the number o f i n t e r n a l degrees o f freedom of the molecule. Although t h i s pro­ blem i s f a r from trivial, algorithms have been developed which allow the more o r l e s s r o u t i n e use o f the computer i n accomplish­ i n g t h i s . These i n c l u d e adaptations o f the not so efficient steepest descent method (2) and f a s t e r Newton-Raphson (3,4,5), Fletcher-Reeves (6) and r e l a x a t i o n (7) methods. Furthermore, the e n e r g e t i c s o f t r a n s i t i o n s between d i f f e r e n t conformations can o f t e n be computed by c o n s t r a i n i n g c e r t a i n i n t e r n a l coordinates t o a succession of f i x e d values which " t r a c k " the t r a n s i t i o n and minimizing the energy a t each step with r e s p e c t t o the other in­ ternal coordinates (8). Thus we may consider the problem of application o f the model t o o r d i n a r y s i z e d organic molecules t o be e s s e n t i a l l y s o l v e d . Further, experience has shown that the model itself is valid f o r s i g n i f i c a n t c l a s s e s of organic (and other) molecules. The question n a t u r a l l y a r i s e s , then, as t o whether it might u s e f u l l y be extended t o bulk matter r a t h e r than j u s t i s o l a t e d molecules. I t turns out that there are some very i n t e r e s t i n g problems i n molecular s o l i d s t h a t can be s t u d i e d by the method. A c t u a l l y , i t might be s a i d t h a t the f i r s t a p p l i c a t i o n of mo­ l e c u l a r mechanics c a l c u l a t i o n s were to c r y s t a l s . C r y s t a l para­ meters and heats of s u b l i m a t i o n o f rare-gas s o l i d s and other sim­ p l e molecules have long been used t o help determine i n t e r a t o m i c and i n t e r m o l e c u l a r p o t e n t i a l s (9). The problem o f c a l c u l a t i n g c r y s t a l s t r u c t u r e s of molecular s o l i d s i n general doesn't r e a l l y

0-8412-0463-2/78/47-086-137$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

138

g r e a t l y extend the computational magnitude because o f the r e ­ s t r a i n t s of the c r y s t a l symmetry. We do not address ourselves to t h i s i n t e r e s t i n g and important a p p l i c a t i o n here. However, the s t r u c t u r e and motion of d e f e c t s i n molecular s o l i d s does r e q u i r e the minimization o f the conformational energy of mechanically s t a b l e and non-regular assemblages of molecules. The mechanical­ l y s t a b l e d e s c r i p t i o n (where a s i n g l e minimum p o t e n t i a l energy c o n f i g u r a t i o n of a l a r g e assembly i s sought) i s used to d i s t i n ­ guish the problem from that of "Monte C a r l o " or "molecular dyna­ mics" c a l c u l a t i o n s (where the o b j e c t i s to evaluate the p o t e n t i a l energy or t r a j e c t o r i e s of a l a r g e number of random c o n f i g u r a t i o n s f o r s t a t i s t i c a l averaging purposes). An important example of the d e s i r a b i l i t y of c a l c u l a t i n g the s t r u c t u r e and m o b i l i t y of d e f e c t s i n molecular s o l i d s l i e s i n the area of polymer c r y s t a l s . The onset of chain m o b i l i t y without melting (which o f t e n can be t r e a t e d as d e f e c t motion) with i n ­ c r e a s i n g temperature r e s u l t c r y s t a l s t i f f n e s s (10) tegy which has been s u c c e s s f u l i n handling the energy minimiza­ t i o n of the l a r g e assembly o f atoms c h a r a c t e r i s t i c of the e n v i ­ ronment of a d e f e c t i n a molecular c r y s t a l . I t was f i r s t a p p l i e d to the p r o p e r t i e s of kinks i n polyethylene (11) and has been sub­ sequently a p p l i e d to other problems (12,13) (see F i g u r e 1) but the methodology has not been d e s c r i b e d i n d e t a i l b e f o r e . A Strategy f o r Energy M i n i m i z a t i o n The conformational energy of a molecular c r y s t a l may ten as the sum o f the i n t r a and i n t e r m o l e c u l a r energies u

= u

i

n

t

r

a

+ U

i n t e r

be w r i t ­

.

(1)

The i n t r a m o l e c u l a r energy i s approximated as the sum of energies of deformation of valence bond lengths R^, valence angles 0^ and t o r s i o n a l angles 0^ and energies of non-bonded (Vander Waals) i n ­ t e r a c t i o n s at d i s t a n c e s R , n

U i n t r a = E ^ C R i - R?)

+ E^k ? (S-

2

+

ZV

(R

NB n>

2

- ©°) +

(1 + cos

'

30 ) k

( 2 )

The i n t e r m o l e c u l a r energy i s the sum of non-bonded energies as­ s o c i a t e d with i n t e r m o l e c u l a r atom d i s t a n c e s , Rm "inter =

™m

<*m> •

3

<>

In the energy minimization of i s o l a t e d organic molecules i t has been found e f f i c i e n t to use a h i g h l y cooperative a l g o r i t h m i n which a t each step of an i t e r a t i v e scheme a l l atoms move coopera­ t i v e l y towards the minimum. T h i s i s e x e m p l i f i e d by multi-dirnen-

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

12.

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Energy Calculations and Polymer Crystals

139

s i o n a l Newton-Raphson methods. The Newton-Raphson Method (4). The energy i n Eq. (2) may be expanded about a t r i a l geometry through q u a d r a t i c terms i n the displacements from the t r i a l s t r u c t u r e AR^, A e j , A 0 and AR . However, t h i s f u n c t i o n i s h i g h l y redundant i n these i n t e r n a l co­ o r d i n a t e s . The redundancy may be removed by expressing each i n ­ t e r n a l coordinate displacement i n terms of the three c a r t e s i a n displacements AXp(a = 1,2,3) of each o f the appropriate atoms, p. The molecular energy becomes through q u a d r a t i c terms i n the Ax's, k

U = ZA AXp + Z^CpqAxgAx^'

n

(4)

p

where the constants Ap, Cp^ now i n v o l v e both the p o t e n t i a l func­ t i o n constants of Eq. (2) and g e o m e t r i c a l f a c t o r s from the t r a n s ­ formations from i n t e r n a l t o c a r t e s i a n c o o r d i n a t e s . Eq. (4) can be d i f f e r e n t i a t e d wit l i n e a r equations. Fro ward the minimum i s obtained that i s used to c a l c u l a t e a new t r i a l structure. I t e r a t i o n i s continued u n t i l the A and the displacements approach zero. While there i s no guarantee o f convergence, i t has been found that w i t h the added feature of damping or a t t e n u a t i n g large displacements the method converges r e l i a b l y i n a p r a c t i c a l number of i t e r a t i o n s (^2-10) and t h a t the number of i t e r a t i o n s tends t o be independent of molecular s i z e . However, the s o l u t i o n of the l i n e a r equations a t each i t e r a t i o n a s s o c i a t e d with t h i s method becomes i m p r a c t i c a l f o r l a r g e systems (>M>0 atoms) because of the dependence of computation time. p

A Hybrid Newton Raphson R e l a x a t i o n Method. A s e c t i o n of one polymer chain c o n t a i n i n g or neighboring a conformational d e f e c t i s about the r i g h t s i z e t o be handled w e l l by the Newton-Raphson method (see Figure 2). T h i s suggests p r e s e r v i n g i t s b a s i c u t i l ­ i t y and t r y i n g a "hybrid" technique. We f i n d t h a t a h y b r i d a l ­ gorithm i n which the i n t e r n a l degrees o f freedom o f each molecule are minimized by Newton-Raphson but i n which the i n t e r a c t i o n s be­ tween molecules are handled by a r e l a x a t i o n method i s e f f i c i e n t . The method proceeds as f o l l o w s : (a) From an i n i t i a l t r i a l s t r u c t u r e the energy (with respect to i t s i n t e r n a l coordinates and i t s p o s i t i o n and o r i e n t a t i o n ) of a s e l e c t e d molecule f r e e to move i n the f i e l d of i t s f i x e d neighbors i s minimized by i t e r a t i v e Newton-Raphson. (b) In t u r n each molecule i n the assembly has i t s energy m i n i ­ mized i n the f i e l d of the other f i x e d molecules. (c) The t o t a l energy o f the assembly i s computed a t the end of each complete c y c l e through the molecules. (d) The c y c l i n g i s repeated u n t i l convergence of the t o t a l ener­ gy i s o b t a i n e d . In Step (a) above i t i s necessary to modify the NewtonRaphson a l g o r i t h m used f o r f r e e i s o l a t e d molecules s l i g h t l y t o

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

140

COMPUTER MODELING OF MATTER

Figure 1. Representative conformation defects in polyethylene chains, hold kink, (b) Reneker twist, and (c) smooth twist.

(a)

(b)

(c)

Figure 2. Crystal packing in polyethylene (a, b projection), one coordination shell about a central chain is shown. Unit cell boundaries also are shown. In defect calculations, center chain (1) is replaced by one containing a conformational defect (see Figure 1). In the hybrid Newton-Raphson relaxation method, one chain at a time has its energy minimized with the other chains fixed. In the free chain, all internal degrees of freedom participate (torsional angles, etc.) as well as the position and orientation of the chain. After a cycle through all of the chains, the total energy of the assembly is computed. The cycles are repeated until the energy is stable.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Energy Calculations and Polymer Crystals

141

accommodate the i n t e r m o l e c u l a r non-bonded i n t e r a c t i o n s i n c r y s ­ tals. In Eq. (2) above, R i n v o l v e s an i n t r a m o l e c u l a r d i s t a n c e between two atoms, both of which are f r e e to move toward the optimum p o s i t i o n . In the r e l a x a t i o n method, Rm (Eq. (3)) repre­ sents an i n t e r m o l e c u l a r i n t e r a c t i o n i n which the atom i n the c h a i n s e l e c t e d f o r minimization moves but the other one i s f i x e d . T h i s s i t u a t i o n r e q u i r e s an a d d i t i o n a l subroutine i n the NewtonRaphson program. In the conventional Newton-Raphson program a p p l i e d to s i n g l e molecules a permanent t a b l e of the i n t e r a c t i o n s contained i n Eq. (2) i s set up and used throughout the c a l c u l a ­ tion. Because of the l a r g e number of i n t e r m o l e c u l a r i n t e r a c t i o n s t h i s i s abandoned i n the i n t e r m o l e c u l a r Rm subroutine. The l i s t of i n t e r a c t i o n s i s generated dynamically i n the R subroutine during each iteration from the l i s t s of atoms coordinates. Due to the l a r g e number of i n t e r m o l e c u l a r i n t e r a c t i o n s com­ p u t a t i o n time i s reduced by employing a d i s t a n c e c u t - o f f beyond which i n t e r a c t i o n s ar f u n c t i o n s f o r carbon an and we employ i n the R subroutine the same c u t - o f f used by him i n developing the f u n c t i o n s . n

m

m

The Banded-Matrix Approximation. The l i n e a r nature of the polymer chains allows an approximation which g r e a t l y speeds up the i n d i v i d u a l Newton-Raphson minimizations. As d i s c u s s e d above, the Newton-Raphson method r e s u l t s i n a s e t of l i n e a r equations (represented by the c o e f f i c i e n t or "C" matrix i n Eq. ( 4 ) ) . The operations i n the Gauss r e d u c t i o n of the system increase as N^ (where N i s the number o f v a r i a b l e s ) . Atoms that are s p a t i a l l y removed from each w i l l have small i n t e r a c t i o n c o e f f i c i e n t s , Cpq. In a l a r g e molecule many of the c o e f f i c i e n t s are s m a l l . In an extended l i n e a r molecule (such as encountered i n the present con­ text) numbering the atoms c o n s e c u t i v e l y from one end to the other ensures t h a t the l a r g e c o e f f i c i e n t s w i l l be near diagnoal (see F i g u r e 3). Those c o e f f i c i e n t s f a r o f f diagonal w i l l represent s p a t i a l l y removed i n t e r a c t i o n s and w i l l be s m a l l . Thus we can approximate the C matrix as a banded one. The number of opera­ t i o n s i n reducing i t i n c r e a s e s as N f o r l a r g e N. T h i s allows handling q u i t e long chain segments i n the c r y s t a l array with reasonable computation times. Computational

Results

The crux of the present method i s how w e l l the r e l a x a t i o n p o r t i o n converges. Some experience has now been gained with i t (11/12,13). F i g u r e 4 shows r e s u l t s of c a l c u l a t i o n s made i n our l a b o r a t o r y (11,13) on the energies of three conformational de­ f e c t s i n polyethylene c r y s t a l s . These are a kink (15, 6) , a Reneker t w i s t (1£) and a smooth t w i s t (13) (see F i g u r e 1). The f i r s t two have been proposed as s t a b l e p o i n t d e f e c t s and the l a s t one as a t r a n s i t i o n s t a t e f o r the motion that accomplishes

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

\

~ \

c,

C„

2

c

\

l 3

0 \

c, 2

C2

C3 C 4

^3

C

2

c, 3

2

C

25

'

N

\

2

42

\

C34 C35 '

C43 c

C63 \

33

4 4

c

C 4 ^ 6

4 5

6

5

c

^

\ 4 6

6

^75 W 6

Figure 4. Convergence of the relaxation method. The energy of a crystalline array of polyethylene chains with the central one containing the defect indicated (relative to a nondefective array), see Figure 1. The Pechhold kink is in an array of two free coordination shells (6 +12 = 18 chains), each chain is C Il o> The other two defects are in an array of one free coordination shell plus a rigid second coordination shell, each chain is J!t

N

\ ^

C

Figure 3. Coefficient matrix in NewtonRaphson method. In a linear (nearly) extended chain, units far apart in numbering are far apart spatially and have small interaction coefficients. This orders the matrix in a banded manner that allows Gauss reduction to be greatly accelerated.

2

\

S

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

6

•

^

6

7

'77

12.

BOYD

Energy Calculations and Polymer Crystals

143

net r o t a t i o n - t r a n s l a t i o n of a c r y s t a l chain. Although the c a l c u ­ l a t i o n s have not been c a r r i e d out to an extreme number of c y c l e s i t appears the convergence i s s a t i s f a c t o r y . The computation times are not excessive. For two f l e x i b l e c o o r d i n a t i o n s h e l l s about a polyethylene kink d e f e c t (19 chains, each C H ) three complete r e l a x a t i o n c y c l e s r e q u i r e d about 30 min. computation time on a Univac 1108 system. Other assemblies should be roughly p r o p o r t i o n a l to the number of chains and the number o f u n i t s w i t h i n each chain. The consequences of the computed d e f e c t p r o p e r t i e s f o r p o l y ­ mer c r y s t a l behavior have been d i s c u s s e d i n the l i t e r a t u r e c i t e d and w i l l not be gone i n t o here. We s u f f i c e to conclude t h a t i t i s p o s s i b l e t o devise a computational s t r a t e g y t h a t allows the energy minimization of c h a i n assemblies l a r g e enought t o permit r e a l i s t i c and p r a c t i c a l s i m u l a t i o n o f the energies and packings of d e f e c t s i n polymer c r y s t a l s . The U. S. Army Researc Science Foundation provide ported here. 1 4

3 Q

Literature Cited 1. 2. 3. 4. 5. 6. 7. 8. 9. 10.

11. 12. 13. 14. 15. 16.

A l l i n g e r , N. L., Adv. Phys. Org. Chem. (1976) 13, 1. Wiberg, K. B., J . Am. Chem. Soc. (1965) 87, 1070. Jacob, E. J . , Thompson, H. and B a r t e l l , L. S., J . Chem. Phys. (1967) 47, 3736. Boyd, R. H., J . Chem. Phys. (1968) 4 9 , 2574. L i f s o n , S. and Warshel, A., J . Chem Phys. (1968) 49, 5116. Scherr, H., Hägele, and Grossman, H. P., C o l l o i d Polym. Sci. (1974) 252, 871. A l l i n g e r , N. L., J . Amer. Chem. Soc. (1971) 93, 1637. Wiberg, K. B. and Boyd, R. H., J . Amer. Chem Soc. (1972) 94, 8426. H i r s c h f e l d e r , J . O., C u r t i s s , C. F. and B i r d , R. B., "Mole­ c u l a r Theory o f Gases and L i q u i d s " Wiley, New York (1954). McCrum, N. G., Read, B. E. and Williams G., " A n e l i s t i c and D i e l e c t r i c E f f e c t s i n Polymeric S o l i d s " Wiley, New York (1967). Boyd, R. H., J . Polym. Sci.-Polym. Phys. Ed. (1975) 13, 2345. Reneker, D. H., Fanconi, B. M. and Mazur, J . , J . Appl. Phys. (1977) 48, 4032. M a n s f i e l d , M. and Boyd, R. H., J . Polym Sci.-Polym. Phys. Ed. (1978) in press. W i l l i a m s , D. E., J . Chem Phys. (1967) 47, 4680. Pechhold, W., Blasenbry, S. and Woerner, C o l l o i d Polym. Sci. (1963) 189, 14. Reneker, D. H., J . Polym. S c i . (1962) 59, 539.

RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13 Multiple Time Step Methods and an Improved Potential Function for Molecular Dynamics Simulations of Molecular Liquids W. B. STREETT and D. J. TILDESLEY 1

2

Science Research Laboratory, United States Military Academy, West Point,NY10996 G. SAVILLE Department of Chemical Engineering, Imperial College of Science, Prince Consort Road, London S.W.7., England Among the importan for molecular dynamic s i n g u l a r i t y free a l g o r i t h m f o r rigid polyatomics, developed by Evans and Murad(1,2),and the m u l t i p l e time step (MTS) method, developed by S t r e e t t , et al. (3). In the method o f Evans and Murad, the equations o f r o t a t i o n a l motion are expressed i n terms o f four parameters, x, n, ξ, ζ, c a l l e d quaternions, def i n e d by X=Cos (6/2) Cos ((!/;+<{>) /2) n=Sin(9/2)Cos((!ji-<|>)/2) S=Sin(6/2)Sin((iJH>)/2) C=Cos(0/2)Sin((i|;+<J>)/2) , where θ, Ψand ф are the E u l e r angles d e f i n e d by G o l d s t e i n (4). They report s i g n i f i c a n t improvements i n accuracy and e f f i c i e n c y over methods based on the equations o f E u l e r , Lagrange o r Ham­ ilton, using E u l e r angles. The MTS method of S t r e e t t , e t al., based on the use o f two o r more time steps o f d i f f e r e n t lengths to i n t e g r a t e the equations o f motion i n systems governed by continuous p o t e n t i a l f u n c t i o n s , has r e s u l t e d i n i n c r e a s e s i n computing speed by f a c t o r s o f three t o eight over conventional methods (3). The o r i g i n a l MTS paper d e s c r i b e s i n d e t a i l how the method i s a p p l i e d t o systems o f s p h e r i c a l molecules. In t h i s paper we give s i m i l a r d e t a i l s o f the a p p l i c a t i o n o f the MTS method t o systems o f rigid polyatomic molecules o f a r b i t r a r y shape, using the a l g o r i t h m o f Evans and Murad. We a l s o

Current address: Chemical Engineering Department, Cornell University, Ithaca, NY 14853. Current address: Physical Chemistry Laboratory, South Parks Road, Oxford OX1 3QZ, England. 1

2

0-8412-0463-2/78/47-086-144$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

Multiple Time Step Methods of Molecular Liquids 145

STREETT ET AL.

d e s c r i b e a modified s i t e - s i t e p o t e n t i a l , the s h i f t e d force po­ t e n t i a l , that gives b e t t e r accuracy and s t a b i l i t y than the usual truncated p o t e n t i a l . S i t e - S i t e P o t e n t i a l s For Molecular

Liquids

Since s i t e - s i t e p o t e n t i a l s (also c a l l e d atom-atom poten­ t i a l s ) have been widely used i n computer s i m u l a t i o n and theo­ r e t i c a l s t u d i e s o f molecular l i q u i d s , we describe the a p p l i c a ­ t i o n o f MTS methods to a model p o t e n t i a l of t h i s type. MTS methods are completely general, and can be a p p l i e d e q u a l l y w e l l to other types o f p o t e n t i a l f u n c t i o n s . In s i t e - s i t e models the p o t e n t i a l energy between a p a i r o f molecules i s the sum o f the p o t e n t i a l energies between p a i r s o f s i t e s on d i f f e r e n t molecules. These s i t e s are u s u a l l y atoms or groups of atoms, which i n t e r ­ a c t with s i t e s on other molecules through s p h e r i c a l l y symmetric p o t e n t i a l f u n c t i o n s . Thu cules 1 and 2 takes th

u

(

r

12

h

)

i 2 ^ u

,

B

u

aB

(

r

aB ' )

where OK i s the set of angles d e f i n i n g the o r i e n t a t i o n o f mole­ cule i , g ( g ) Is the p o t e n t i a l energy between s i t e a on molecule 1 and s i t e 6 on molecule 2 (a f u n c t i o n only o f the s i t e - s i t e distance and the summation i s over a l l s i t e s i n each molecule. The methods d e s c r i b e d h e r e i n are a p p l i c a b l e to models governed by s i t e - s i t e p o t e n t i a l s that are continuous functions of r . Truncated P o t e n t i a l s . In the i n t e r e s t o f computing e f f i c ­ iency, s i t e - s i t e p o t e n t i a l s are truncated at a c u t o f f d i s t a n c e , r , u s u a l l y taken to be about 2.5 times the e f f e c t i v e "diameter" o f the s i t e . I n t e r a c t i o n s beyond t h i s d i s t a n c e are neglected, and c o r r e c t i o n s to the c a l c u l a t e d thermodynamic p r o p e r t i e s , due to the neglected " t a i l " , are added using a simple p e r t u r b a t i o n scheme (_5) . E r r o r s inherent i n the use of truncated p o t e n t i a l s i n molecular dynamics simulations have u s u a l l y been overlooked, or at l e a s t neglected. These e r r o r s a r i s e because of the im­ pulse f e l t by two molecules whose s e p a r a t i o n r crosses the boundary r=r i n s u c c e s s i v e time s t e p s . There i s a gain or l o s s of p o t e n t i a l energy that i s not balanced by an equal and opposite change i n k i n e t i c energy; t o t a l energy i s not conser­ ved, and the c a l c u l a t e d t r a j e c t o r i e s depart s l i g h t l y from true Newtonian t r a j e c t o r i e s . (Numerical round o f f causes s i m i l a r errors.) In simulations of homogeneous f l u i d s there are, on average, as many c r o s s i n g s of the boundary i n one d i r e c t i o n as the other, so there i s no appreciable d r i f t i n the t o t a l energy and no s i g n i f i c a n t e r r o r i n the c a l c u l a t e d e q u i l i b r i u m p r o p e r t i e s ; however, i n simulations o f non-homogeneous systems (e.g., g a s - l i q u i d i n t e r f a c e ) these impulses can be a s i g n i f i u

r

a

a

r

c

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

146

COMPUTER MODELING OF MATTER

cant source o f e r r o r and i n s t a b i l i t y . In a d d i t i o n , the accum­ u l a t i o n o f small e r r o r s i n p a r t i c l e t r a j e c t o r i e s i s l i k e l y t o r e s u l t i n s i g n i f i c a n t e r r o r s i n c a l c u l a t i o n s o f time-dependent phenomena that are based on c o r r e l a t i o n s o f p r o p e r t i e s a t long times, such as the "long time t a i l s " o f the v e l o c i t y autocor­ r e l a t i o n f u n c t i o n (6^) . For reasons t o be made c l e a r i n the next s e c t i o n , truncated p o t e n t i a l s have an adverse e f f e c t on energy c o n s e r v a t i o n i n some MTS methods. These problems can be overcome by simply i n c r e a s i n g the c u t o f f distance r t o a p o i n t where the i n t e r a c t i o n s omitted are t r u l y n e g l i g i b l e ; how­ ever, s i n c e the number o f p a i r i n t e r a c t i o n s c a l c u l a t e d i n the s i m u l a t i o n increases approximately as r^,, t h i s remedy i s ex­ pensive. As an a l t e r n a t i v e , we have developed a modified poten­ t i a l , c a l l e d the s h i f t e d force p o t e n t i a l , that provides accep­ t a b l e accuracy and energy conservation without increases i n r . The S h i f t e d Force P o t e n t i a l When two s i t e s i n t e r a c t through a s p h e r i c a l l y symmetri upon each other equal an vector £ between them. (A wavy underline i n d i c a t e s a vector quantity.) The magnitude o f the force i s given by F=-(du/dr). In Figure 1(a) the p o t e n t i a l energy and force between two s i t e s are p l o t t e d against distance f o r a t y p i c a l truncated p o t e n t i a l . I t i s the d i s c o n t i n u i t y i n the force at r = r that i s the source of e r r o r i n molecular dynamics s i m u l a t i o n s . To reduce the e r r o r we use a m o d i f i e d f o r c e , F , formed by s u b t r a c t i n g from the usual e x p r e s s i o n f o r the force a constant, H, equal to the mag­ nitude o f t h i s d i s c o n t i n u i t y . The modified force i s c

c

c

m

F =-(du/dr)-H,

rr.

where H=-(du/dr) The

r=r m o d i f i e d p o t e n t i a l i s defined

i n terms o f t h i s f o r c e : (2)

This

gives u(r)-Hr+G,

r
(2a)

%
r>r

where G i s a constant given by G=Hr -u(r ). c

c

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

STREETT ET AL.

Multiple Time Step Methods of Molecular Liquids 147

The modified force and p o t e n t i a l energy are p l o t t e d i n Figure 1(b). Both p o t e n t i a l energy and force now go smoothly t o zero at r = r . The m o d i f i c a t i o n to the force i s a constant equal t o the magnitude o f the o r i g i n a l d i s c o n t i n u i t y a t r , hence the name s h i f t e d force p o t e n t i a l . The m o d i f i c a t i o n t o the poten­ t i a l energy i s a s t r a i g h t l i n e u (r)=-Hr+G, with slope -H and with u and r i n t e r c e p t s G and r , r e s p e c t i v e l y . These m o d i f i ­ c a t i o n s are a p p l i e d only f o r r
Q

Q

c

c

M u l t i p l e Time Step Methods S p h e r i c a l Molecules. Before d e s c r i b i n g the a p p l i c a t i o n o f MTS methods t o molecular l i q u i d s , we o u t l i n e b r i f e l y how they are a p p l i e d t o s p h e r i c a l molecules; complete d e t a i l s can be found i n the o r i g i n a l paper (3) . The essence o f the method i s the separation o f the t o t a l instantaneous force a c t i n g on each molecule i n t o a primary f o r c e , Fp, produced by a cage o f near­ est neighbors, and a secondary f o r c e , F , produced by the r e ­ maining neighbors w i t h i n the c u t o f f d i s t a n c e , and the use o f time steps o f d i f f e r e n t lengths to c a l c u l a t e the time e v o l u t i o n of these f o r c e s . In general, Fp>F , and s i m i l a r i n e q u a l i t i e s h o l d f o r time d e r i v a t i v e s o f these f o r c e s . This means that the motion o f a molecule i s dominated by a strong, r a p i d l y v a r y i n g primary f o r c e , produced by a cage o f nearest neighbors, while the smaller secondary force changes more slowly with time. In conventional molecular dynamics methods, no d i s t i n c t i o n i s made between primary and secondary f o r c e s ; the forces between a l l p a i r s w i t h i n the c u t o f f d i s t a n c e are r e c a l c u l a t e d every time step, and i t i s t h i s summation over i n t e r a c t i n g p a i r s that consumes most o f the computing time. In the MTS method, only the primary forces are r e c a l c u l a t e d a t every step, while time steps f i v e t o twenty times longer are used t o p r e d i c t the time e v o l u t i o n o f secondary f o r c e s . In general, the number o f p r i s

s

American Chemical Society Library

1155 16th St., N.W. Washington, 20036 In Computer Modeling ofO.C. Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

148

COMPUTER MODELING OF MATTER

o

(a)

(b)

Figure 1. Potential energy u (solid lines) and force F (dashed lines) as a function of site-site distance r for truncated (a) and shifted-force (b) potential functions. The modified force in (b) is formed by shifting the force-distance curve in (a) upward a distance H, equal to the magnitude of the discontinuity at r = r . The modified potential energy in (b) is then defined by Equations 1 and 2. c

Figure 2. The vector r from site (3 of molecule 2 to site a of molecule 1, in the space-fixed axis system, is given by Equation 6. Open circles are molecular centers. To apply the Taylor series MTS methods (Equation 3), one or more time derivatives of r must be calculated.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

STREETT ET AL.

Multiple Time Step Methods of Molecular Liquids

Figure 3. Diagram illustrating the conventions chosen for excluding interactions beyond the cutoff distance and for distinguishing between primary and secondary forces. Open circles are molecular centers and solid circles are additional sites. All sites on molecules 2, 2, and 3 contribute to the primary forces on the central molecule A, because each has at least one site within a distance r„ from a site of A. Molecules 4, 5, and 6 do not interact with A because their centers lie beyond the cutoff distance, r , from the center of A. All site-site interactions between molecule A and the remaining (unlabeled) molecules contribute to the secondary forces on A. r

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

150

COMPUTER MODELING OF MATTER

mary neighbors i s about an order o f magnitude l e s s than the num­ ber w i t h i n the c u t o f f d i s t a n c e , so the average number o f p a i r i n t e r a c t i o n s e x p l i c i t l y c a l c u l a t e d a t each time step i s g r e a t l y reduced. At time tQ, the primary and secondary f o r c e s , Fp(tQ) and F ( t Q ) , are c a l c u l a t e d f o r each molecule. Concurrently, the time d e r i v a t i v e s , F (tg) , F s ' ^ o ) ' " * ' o f the secondary f o r c e are c a l c u l a t e d , and l i s t s o f the primary neighbors are compiled for each molecule. At each o f the next n-1 time steps (where, t y p i c a l l y , n=10) the primary forces are r e c a l c u l a t e d i n the usual way, u s i n g the l i s t o f primary neighbors compiled a t t g , while the secondary forces a r e c a l c u l a t e d from a T a y l o r s e r i e s , using time d e r i v a t i v e s c a l c u l a t e d a t t g : s

-

s

F (1 +kA t) =F ( t ) +F ( t ) — + . . . + F J s

0

S

0

S

m )

0

(to) l i ^ i l + m

;

Here A t i s the time step, and we s h a l l r e f e r t o the T a y l o r s e r i e s t r u n c a t e d a f t e r term m as the mth order T a y l o r s e r i e s . An a l t e r n a t i v e f i r s t order p r e d i c t i o n scheme, based on the f i n ­ i t e d i f f e r e n c e formula , gs(to+At)-F^^ ~s At F

=

(4)

has a l s o been s u c c e s s f u l l y t e s t e d . In t h i s method, which we c a l l the l i n e a r e x t r a p o l a t i o n MTS method, the primary and secondary forces between a l l p a i r s w i t h i n the c u t o f f d i s t a n c e are c a l ­ c u l a t e d a t each o f the f i r s t two steps i n each b l o c k o f n s t e p s , and a l i n e a r e x t r a p o l a t i o n o f the secondary forces i s c a r r i e d out over the next n-2 s t e p s . When the l i n e a r e x t r a p o l a t i o n method i s a p p l i e d t o molecular l i q u i d s , torques must be sepa­ r a t e d i n t o primary and secondary components and t r e a t e d i n the same way as f o r c e s . This method i s s i m i l a r t o the f i r s t order T a y l o r s e r i e s method (equation 3 with two terms), b u t i t i s simpler to use and r e q u i r e s l e s s storage. When a t r u n c a t e d p o t e n t i a l i s used, the l i n e a r e x t r a p o l a t i o n method gives b e t t e r energy c o n s e r v a t i o n because i t i n c l u d e s the e f f e c t s o f molec­ u l a r p a i r s c r o s s i n g the boundary r = r between the f i r s t two time steps i n each b l o c k , and uses t h i s i n f o r m a t i o n i n the ex­ t r a p o l a t i o n o f secondary forces over the next n-2 s t e p s . T a y l o r s e r i e s methods, on the other hand, i n c l u d e the i n h e r e n t assump­ t i o n that no c r o s s i n g s occur i n each b l o c k o f n s t e p s . The use of s h i f t e d f o r c e p o t e n t i a l s with these methods r e s u l t s i n im­ proved accuracy and s t a b i l i t y . Comparisons o f computing speeds and storage requirements f o r v a r i o u s MTS methods a r e given i n the next s e c t i o n . Expressions f o r the time d e r i v a t i v e s F , F g , * * - , i n c

1

s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

STREETT ET AL.

equation

Multiple Time Step Methods of Molecular Liquids

(3) have been published

(3) .

151

The f i r s t o f these i s

1

F =Ar'+B(r•r')r,

(5)

where A and B are functions only o f the s c a l a r r . In equation (5) the time dependence o f the force between a p a i r o f mole­ cules i s thus contained i n the time d e r i v a t i v e r ' o f the vec­ t o r separating them. Higher d e r i v a t i v e s o f F contain higher d e r i v a t i v e s o f r . The vector r - ^ between two s p h e r i c a l mole­ cules i s a simple vector sum: E i 2 £ l ~ H 2 ' ^ L± ^ p o s i t i o n vector o f molecule i i n a space-fixed frame. This expression can be d i f f e r e n t i a t e d t o give £ i 2 £ i ~ £ 2 ' S i 2 ^ i ' " £ 2 ' ' e t c . The time d e r i v a t i v e s o f r - ^ are simple vector sums o f the v e l o c i t i e s , a c c e l e r a t i o n s , and higher time d e r i v a t i v e s o f the center o f mass p o s i t i o n s . Molecular L i q u i d s . In the case o f r i g i d polyatomic mole­ cules governed by s i t e - s i t l y more complicated, becaus the center o f mass, and r o t a t i o n a l motion must be taken i n t o account. In most schemes f o r s o l v i n g the equations o f motion of polyatomics, the motion i s separated i n t o t r a n s l a t i o n o f the center o f mass and r o t a t i o n about an axis through the cen­ t e r o f mass. In the summation o f forces step i n the s i m u l a t i o n the t o t a l force a c t i n g on each s i t e o f each molecule i s c a l c u ­ l a t e d by a summation over a l l molecular p a i r s w i t h i n the c u t o f f distance. Once these forces are known, the torque on each molecule i s c a l c u l a t e d from a sum o f vector cross products of the form =

w

e r e

s

t

=

n

e

=

(5a)

where T i s the torque, i s the vector from the center o f mass to s i t e a , and the force on s i t e a . For s i t e - s i t e models, the MTS method i s a p p l i e d only t o the c a l c u l a t i o n o f s i t e - s i t e f o r c e s ; the torques are then c a l c u l a t e d i n the usual way. Figure 2 shows the r e l a t i o n between the s i t e - s i t e vector r connecting s i t e s a and B , the p o s i t i o n vectors R-^ and ^ °f molecules 1 and 2 , and the body-fixed s i t e vectors h and hg. The vector r can be expressed as a

(6)

£ Si-5 &x-fcB • =

Differentiating

2

+

gives r'^-R^+h^-hg, (7) s ^ B i ' - B ^ + b d ' - M ' ' etc.

To apply the MTS method t o molecular l i q u i d s we need t o c a l c u ­ l a t e the d e r i v a t i v e s on the r i g h t side o f equation (7). The

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

152

COMPUTER MODELING OF MATTER

m

d e r i v a t i v e s g ( ) o f the center o f mass p o s i t i o n vectors are r o u t i n e l y c a l c u l a t e d t o order m=4 o r 5 i n molecular dynamics programs based on p r e d i c t o r - c o r r e c t o r algorithms o f the type due t o Gear (7^. To c a l c u l a t e the d e r i v a t i v e s h , h ^ , e t c . , we take advantage o f the elegant features o f the method o f qua­ ternions. In t h i s method, the equations o f r o t a t i o n a l motion are solved i n a body-fixed system o f axes (the p r i n c i p a l a x i s system). Any vector V i n the p r i n c i p a l axis system i s t r a n s ­ formed i n t o a space-fixed v e c t o r by the transformation 1

a

p

yA=s-Yp, where E i s the r o t a t i o n matrix -S +n -c + 2Ux-Sn) 2

E=

Hence the v e c t o r

h

a

2

2

x

-2(£n+Cx) ^-n -c +x

2

2

2

2

2(1U-SX) -2(^+nx)

(8)

i s given by

ha=E»ba, where ka i s the vector from the center o f mass o f the molecule to s i t e a, i n the p r i n c i p a l a x i s system. Now b i s time i n v a r ­ i a n t , so the d e r i v a t i v e s o f ha are given by a

f

hd=E -b , a

(9) f

feT=E

etc.

The d e r i v a t i v e s , £'' , e t c . are obtained from term by term d i f f e r e n t i a t i o n o f equation ( 8 ) . The f i r s t d e r i v a t i v e s o f the quaternions can be c a l c u l a t e d from the p r i n c i p a l angular v e l o c i t y v e c t o r , w, i n the p r e d i c t o r step, using the f o l l o w i n g expression,

V

-x

rT

X n

V

-n

X'

5

n

<

(10)

n

X

X

0

Higher d e r i v a t i v e s can be obtained by successive d i f f e r e n t i a ­ t i o n o f equation (10), using the d e r i v a t i v e s oj , a) , e t c . , which are r o u t i n e l y c a l c u l a t e d i n p r e d i c t o r - c o r r e c t o r i n t e g r a ­ t i o n schemes. Thus equations (8)-(10) provide the means t o c a l ­ c u l a t e the d e r i v a t i v e s 1^, 1 ^ , e t c . , i n equation (7). Once the d e r i v a t i v e s r , r , e t c . , f o r the s i t e - s i t e distances are known, the MTS method i s a p p l i e d t o c a l c u l a t i o n o f the forces i n much the same manner as i n the case o f s p h e r i c a l molecules. 1

11

1

1

1

1

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

Multiple Time Step Methods of Molecular Liquids 153

STREETT ET AL.

C e r t a i n thermodynamic p r o p e r t i e s , such as pressure and i n t e r n a l energy, must be separated i n t o c o n t r i b u t i o n s due t o primary and secondary i n t e r a c t i o n s , and c a l c u l a t e d i n a manner analogous t o t h a t used f o r the forces (_3) . See appendix f o r o u t l i n e o f the molecular dynamics program. Computing Time and Storage

Requirements

General. The r e s u l t s obtained f o r the t e t r a h e d r a l model used here are only a s i n g l e example o f the a p p l i c a t i o n o f MTS methods t o molecular l i q u i d s . Results obtained i n other cases w i l l depend on many f a c t o r s , i n c l u d i n g molecular shape, type of i n t e r m o l e c u l a r p o t e n t i a l f u n c t i o n , and numerical method used to s o l v e the equations o f motion. We d e s c r i b e here i n c r e a s e s i n computing speed and storage requirements a s s o c i a t e d with adapting a F o r t r a n program f o r 108 methane (CH4) molecule i c s to s e v e r a l v a r i a t i o n uses the s i n g u l a r i t y f r e e a l g o r i t h m o f Evans and Murad (2), and a s i t e - s i t e p o t e n t i a l o f the form 6

u(r)=Ar~ +Bexp(Cr), where A, B, C are constants which depend on the type o f s i t e s i t e i n t e r a c t i o n (carbon-carbon, carbon-hydrogen, o r hydrogenhydrogen) . Each molecule has f i v e s i t e s (atoms), so there are 25 s i t e - s i t e i n t e r a c t i o n s per p a i r o f molecules. Distances are expressed i n u n i t s o f a, the e f f e c t i v e "diameter" o f a methane molecule (4. i n t h i s c a s e ) . For the c u t o f f r we use the d i s t a n c e between molecular c e n t e r s : a l l i n t e r a c t i o n s between s i t e s o f a p a i r o f molecules are c a l c u l a t e d i f R i 2 - c ' R-L2 To d i s t i n g u i s h between primary and secondary forces we use the minimum s i t e - s i t e d i s t a n c e , r • , between a p a i r o f mol' min' ^ e c u l e s : i f r •„
r

n

o

n

e

>r

c

III JLII

0

0.

cl

s i t e i n t e r a c t i o n s f o r that p a i r c o n t r i b u t e t o primary f o r c e s ; i f r ^ > r , and R i 2 - c ' H s i t e - s i t e i n t e r a c t i o n s c o n t r i b u t e to secondary f o r c e s . These conventions are i l l u s t r a t e d i n F i g u r e 3. Computer Storage. Increases i n computer storage t o accom­ odate MTS methods are given i n u n i t s o f the number o f molecules, N, and the number o f s i t e s , N , i n the system (e.g., i n a system o f N=108 methane molecules, N =540). In every case the storage r e q u i r e d f o r the l i s t o f primary neighbors i s ^6N. For MTS methods based on the T a y l o r s e r i e s expansion, equation ( 3 ) , there i s an added requirement o f ^5N words f o r each term i n equation (3). Thus f o r a 2nd order T a y l o r s e r i e s (three terms) the storage i n c r e a s e s by ^15N . The l i n e a r e x t r a p o l a t i o n method (equation 4) r e q u i r e s only about 25N a d d i t i o n a l words o f s t o r a g e . Examples o f storage requirements f o r s e v e r a l MTS programs are given i n the t a b l e below. Computing Time. In molecular dynamics s i m u l a t i o n s most o f r

m

n

a

a

s

s

S

S

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

154

COMPUTER MODELING OF MATTER

the computing time i s spent i n c a l c u l a t i n g i n t e r a c t i o n s between p a i r s o f molecules that l i e w i t h i n a s p e c i f i e d c u t o f f d i s t a n c e , hence computing speed v a r i e s i n v e r s e l y with the average number o f p a i r i n t e r a c t i o n s c a l c u l a t e d per time step. Let us c o n s i d e r as a base f o r comparison a conventional molecular dynamics pro­ gram i n which t h i s average number i s N . Using MTS methods, the s i m u l a t i o n i s c a r r i e d forward i n blocks o f n steps where, during the f i r s t one or two steps the usual N i n t e r a c t i o n s are c a l c u l a t e d , together with the d e r i v a t i v e s i n equation (3) or (4), while during the remaining steps the i n t e r a c t i o n s between a l e s s e r number o f primary neighbors, Np, are c a l c u l a t e d . N i s determined by the c u t o f f d i s t a n c e r , and Np by the d i s t a n c e r used to d i s t i n g u i s h between primary and secondary neighbors. For the t e t r a h e d r a l model d e s c r i b e d here we have used a
t

t

c

a

a

c

t

a

1 5

Program Conventional, without MTS MTS, l i n e a r e x t r a p o l a t i o n MTS, 2nd order Taylor s e r i e s MTS, 2nd order T a y l o r s e r i e s MTS, 2nd order T a y l o r s e r i e s MTS, 3rd order T a y l o r s e r i e s MTS, 3rd order T a y l o r s e r i e s

r

n

a

—

—

1.3 1.1 1.2 1.2 1.0 1.2

6 6 10 12 6 15

Storage Computing (10 words) Speed 3

30 33 38 38 38 41 41

1 2.4 2.6 3.2 3.2 2.5 3.4

These f i g u r e s show that the MTS method increases computational speeds by f a c t o r s o f three or more when a p p l i e d to a t e t r a ­ h e d r a l atom-atom model. Evans and Murad (2) r e p o r t comparable

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

STREETT ET AL.

Multiple Time Step Methods of Molecular Liquids

i n c r e a s e s f o r the s i n g u l a r i t y f r e e algorithm over previous methods based on other algorithms; hence the combination o f MTS methods with t h e i r a l g o r i t h m can be expected to increase computing speeds by about an order of magnitude. Our experience with simulations o f diatomic molecules suggests that the advantage i n computing speed of MTS methods i n c r e a s e s with the asymmetry of the molecular model. I t i s t h e r e f o r e l i k e l y t h a t g r e a t e r r e l a t i v e increases i n speed can be r e a l i z e d i n s i m u l a t i o n s o f systems of more h i g h l y asymmetric molecules than those d e s c r i b e d here. In a d d i t i o n , the r e l a t i v e increase i n speed can be expected to i n c r e a s e as the number o f molecules i n the sample i n c r e a s e s . We have demonstrated the power o f MTS methods i n molecular dynamics simulations of both monatomic and polyatomic models. In another r e p o r t i n t h i s volume, J . M. H a i l e d e s c r i b e s how MTS methods have been used to develop an e f f i c i e n t a l g o r i t h m f o r c a l c u l a t i n g three-body We are indebted t d i s c u s s i o n s o f the s i n g u l a r i t y f r e e a l g o r i t h m and the use o f quaternions i n molecular dynamics s i m u l a t i o n s . We acknowledge a generous grant of computing time from the Academic Computer Center of the United States M i l i t a r y Academy, West P o i n t , New York, and f i n a n c i a l support from NSF grant ENG 76-82101. Appendix The f o l l o w i n g o u t l i n e d e s c r i b e s how MTS methods are com­ bined with p r e d i c t o r - c o r r e c t o r i n t e g r a t i o n schemes i n a molec­ u l a r dynamics program based on the s i n g u l a r i t y f r e e a l g o r i t h m of Evans and Murad (2). We consider a system of N polyatomic molecules governed by a s i t e - s i t e p o t e n t i a l f u n c t i o n , i n t e g r a ­ ted by a p r e d i c t o r - c o r r e c t o r method o f order I, using a T a y l o r s e r i e s MTS method o f order m i n which the s i m u l a t i o n i s c a r r i e d forward i n blocks of n time steps. T y p i c a l values of these parameters are: N=108, 256, 500, , £=4 or 5, m=2 or 3, n=10 to 15. The major s e c t i o n s o f the program are as f o l l o w s . 1. I n i t i a l S e c t i o n . Assign i n i t i a l p o s i t i o n s and v e l o c i t i e s to a l l molecules. The p o s i t i o n of a molecule i s determined by i t s center o f mass p o s i t i o n v e c t o r R and four quaternions (x/ n , Ki C ) . We have found i t convenient to use a face centered cubic l a t t i c e , with random molecular o r i e n t a t i o n , as the i n i ­ t i a l configuration. (Note that the quaternions are subject to the c o n s t r a i n t x n + £ + C = l . ) Assign i n i t i a l t r a n s ­ l a t i o n a l v e l o c i t i e s R and body-fixed r o t a t i o n a l v e l o c i t i e s a); the d i r e c t i o n s of these v e c t o r s can be chosen randomly, but the d i s t r i b u t i o n of magnitudes should correspond to the d e s i r e d temperature of the s i m u l a t i o n . C a l c u l a t e f i r s t de­ r i v a t i v e s o f the quaternions from equation (10). Calcu­ l a t e the d e r i v a t i v e s R'' and u' from the forces and torques f o r the i n i t i a l c o n f i g u r a t i o n , using the d i f f e r e n t i a l equa2 +

2

2

2

1

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

155

COMPUTER MODELING OF MATTER

t i o n s o f motion (equations A2), as i n s e c t i o n 3. Set higher d e r i v a t i v e s o f R, u) and the quaternions (to order 2,) equal t o zero. Predictor Section. C a l c u l a t e the f i r s t d e r i v a t i v e s o f the qua­ t e r n i o n s from equation (10), and higher d e r i v a t i v e s (to order I) from repeated d i f f e r e n t i a t i o n o f that equation. Using a T a y l o r s e r i e s o f order I, p r e d i c t values o f R, w and the quaternions, and the f i r s t & d e r i v a t i v e s o f these q u a n t i ­ t i e s , a t time t+At according t o ^JJ"

R(t+At)=R(t)+R' (t)At+-.-+R<*> (

R» (t4At)-R" (t)+R' " (t)4ff...+R ^ (t)(Al)

R^) (t+At)*R<* (This Taylor s e r i e s should not be confused with equation (3), which i s used i n s e c t i o n 3.) Apply the c o n s t r a i n t x l +£ + C = l , t o the p r e d i c t e d quaternions, r e s c a l i n g each one by the f a c t o r 1/(X +TI +S +C ) Evaluation Section. Using p r e d i c t e d p o s i t i o n s and o r i e n t a ­ t i o n s , c a l c u l a t e the t o t a l force on each s i t e o f each mole­ cule. Depending on whether t h i s time step i s the f i r s t time step (a) o r a subsequent time step (b) i n a block o f n steps, proceed as follows with the MTS method. (a) F i r s t Time Step. C a l c u l a t e forces by a summation over a l l p a i r s w i t h i n the c u t o f f d i s t a n c e r , and separate the force on each s i t e i n t o primary (Fp) and secondary ( F ) components (see Figure 3). C a l c u l a t e the d e r i v a t i v e s F^, Si ^ o u t l i n e d i n the main t e x t . For each mole­ c u l e i compile a l i s t o f the remaining molecules that c o n t r i b u t e t o the primary forces on i . (Note that t h i s l i s t need only i n c l u d e molecules j f o r j > i , s i n c e the i - j and j - i i n t e r a c t i o n s are i d e n t i c a l . ) Separate c o n t r i b u t i o n s t o the thermodynamic p r o p e r t i e s (pressure, energy, etc.) i n t o primary and secondary components, and c a l c u l a t e the time d e r i v a t i v e s o f the l a t t e r , using the expressions given i n reference 3. (b) Subsequent Time Step. C a l c u l a t e the primary force on each s i t e and the primary-neighbor c o n t r i b u t i o n s t o the thermodynamic p r o p e r t i e s by a summation over a l l p a i r s i n c l u d e d i n the l i s t o f primary neighbors compiled i n the f i r s t time step. C a l c u l a t e secondary f o r c e s from equation (3) and secondary-neighbor c o n t r i b u t i o n s t o the thermodynamic p r o p e r t i e s from s i m i l a r equations (see reference 3 ) . Add the primary and secondary components o f the c o n t r i b u t i o n s 2 + r

2

2

2

2

2

2

2

c

s

m

a

s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

13.

Multiple Time Step Methods of Molecular Liquids

STREETT ET AL.

157

to the thermodynamic p r o p e r t i e s . Add the primary and secon­ dary forces f o r each s i t e , and c a l c u l a t e the exact f o r c e and torque, i n the s p a c e - f i x e d frame, on each molecule. Trans­ form s p a c e - f i x e d torques, T, i n t o torques i n the p r i n c i p a l axis system, T , using Jp=g~l.T, where E " i s the i n v e r s e o f the matrix i n equation (8). (The matrix i s orthogonal.) C a l c u l a t e the exact t r a n s l a t i o n a l and r o t a t i o n a l a c c e l e r a ­ t i o n s , R' and w', from the d i f f e r e n t i a l equations o f motion: 1

p

1

R' '=F/M (A2) u)'=T / I p

y

y

y=x,y,z,

,

P

y

where M i s the molecular mass and Ip , Ip , I p the p r i n c i p a l moments o f i n e r t i a (4). C a l c u l a t e exact Values o f the f i r s t d e r i v a t i v e s o f the quaternions from equation (10) 4. C o r r e c t o r S e c t i o n . Correc and t h e i r d e r i v a t i v e s ternions (but not t h e i r d e r i v a t i v e s ) , according to z

l(j> =R(J) + -corr ~pred ( . =(J

» C

°

r

. .. ° ^+

r

P

=

r

e

A t )

j

j=0,I,---,*,

'

/ j 8

1

A. *Ao) ,

j = 0 , l , ( A 3 )

(At)Vj!

d

^corr ^pred

A • • AR' ' ~ ,

+

e

'

t

c

"

At/1! 1 1

1

where AR , Aa) and A£' are the d i f f e r e n c e s between the pre­ d i c t e d values from s e c t i o n 2 and the exact values from sec­ t i o n 3. The constants Aj f o r the p r e d i c t o r - c o r r e c t o r method can be found i n t a b l e 9.1 o f reference 7. 5. Return to P r e d i c t o r S e c t i o n . To use the l i n e a r e x t r a p o l a t i o n MTS method, the same pro­ cedure i s followed, except that i n the second time step o f each block o f n steps, the primary and secondary f o r c e s are again c a l c u l a t e d s e p a r a t e l y , using the l i s t o f primary neighbors com­ p i l e d i n the f i r s t time step, and values o f F from equation (4) are used i n a f i r s t order T a y l o r s e r i e s (equation 3) during the next n-2 time steps. I f the torques a r e separated i n t o p r i ­ mary and secondary components, and t r e a t e d i n the same manner as the f o r c e s , equations (3) and (4) can be a p p l i e d t o the t o t a l f o r c e on each molecule, r a t h e r than t o the force on each s i t e , r e s u l t i n g i n a s u b s t a n t i a l savings i n storage. (A s i m i l a r savings i n storage can be r e a l i z e d i n the T a y l o r s e r i e s MTS methods i f the torques are separated i n t o primary and secondary s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

158

COMPUTER MODELING OF MATTER

components, T and T , and d e r i v a t i v e s of T c a l c u l a t e d from d e r i v a t i v e s o f equation (5a); however, the savings i n storage w i l l be o f f s e t by the i n c r e a s e i n computing time r e q u i r e d to evaluate these d e r i v a t i v e s . In t h i s case i t i s more e f f i c i e n t to sum the forces on the s i t e s i n a double loop over a l l r e l e ­ vant molecular p a i r s , and then to c a l c u l a t e the torques from equation (5a) i n a s i n g l e loop over a l l molecules.) g

g

Literature Cited 1. 2. 3.

Evans, D. J . , Molec. Phys. (1977) 34, 317. Evans, D. J . & Murad, S., Molec. Phys. (1977) 34, 327. S t r e e t t , W. B., T i l d e s l e y , D. J . and S a v i l l e , G., Molec. Phys. (1978) in p r e s s . 4. G o l d s t e i n , H., " C l a s s i c a l Mechanics", Chaps. 4 & 5, AddisonWesley, Reading, Massachusetts, 1971. 5. McDonald, I. R. & Singer 43, 40. 6. A l d e r , B. J . & Wainwright, T. E., Phys. Rev. (1970) Al, 18. 7. Gear, C. W., "Numerical Initial Value Problems i n Ordinary Differential Equations," pp. 148-54, P r e n t i c e - H a l l , Englewood Cliffs, New J e r s e y , 1971. RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

14 Optimization of Sampling Algorithms in Monte Carlo Calculations on Fluids JOHN C. OWICKI Department of Chemistry, Stanford University, Stanford, CA 94305

The analysis of simple model classical fluids by Monte Carlo (MC) techniques now is f a i r l complicated systems, where computational expense may limit the precision of the results or the number of calculations performed. Examples include liquid water, dilute solutions, and systems with phase boundaries. Even in "simple" systems, some properties (notably free energies) are very d i f f i c u l t to calculate precisely. The scope of feasible calculations can be expanded by reducing the cost of computing. Since computer technology continues to improve, this factor is significant. MC standard errors are asymptotically proportional to the reciprocal of the square root of the length of the computation, however, so improvements in precision will not be as great as drops in computational costs. Another approach is to improve sampling procedures, since i t is a maxim for MC calculations in general that a properly designed sampling algorithm can reduce s t a t i s t i c a l errors dramatically (1). Therefore, this chapter deals with the optimization of sampling algorithms for MC calculations on bulk molecular systems, p a r t i cularly fluids. Part I is a discussion of some developments in the mathematical theory of MC sampling using Markov chains. The scope is purposely limited, and important topics such as umbrella sampling and sampling in quantum mechanical systems are given short s h r i f t , because they are well-reviewed elsewhere (2.,.3). Nor do we discuss the work of Bennett (4), which deals primarily with e f f i c i e n t data analysis in MC free energy calculations, not with the sampling algorithms per se. Part II of the chapter is a generalization of a sampling algorithm (5) which we have developed for MC calculations on dilute solutions, with some calculations to illustrate i t s usefulness. I.

Theoretical

Developments

The Metropolis Algorithm. For reference purposes and to establish notation, the standard sampling algorithm of Metropolis 0-8412-0463-2/78/47-086-159$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

160

COMPUTER MODELING OF MATTER

et^ a]_. (6) is summarized below. We assume that the reader is already f a i r l y familiar with i t . Underlining denotes a vector or matrix, and, for typographical convenience, subscripts are often indicated in parentheses. The system is assumed to have a large but f i n i t e number of states labeled i = l , 2 , . . . , S , corresponding to different molecular configurations. States occur according to the probability distribution vector TT, where the i t h . element TT(i) is the normalized Boltzmann factor for state i . Transitions between states are made according to the stochastic transition matrix P_, which must satisfy the eigenvalue equation (1)

TT = TTP

There are other conditions on J\ such as irreducibility (1_), but we will not discuss them here. The matrix element p ( i , j ) is the probability of moving t system currently is in stat The Metropolis choice of is p ( i , j ) = q ( i , j ) - m i n [ l , Tr(j)/ir(i)] = 1 - E p(i,k)

i t j i = j

(2)

The elements q ( i , j ) of Q. are defined by the way t r i a l configurations j are generated. Usually one of the N molecules is selected at random, and each of its coordinates is given a perturbation uniformly on some fixed interval [-A,A] (for a spherically symmetric molecule). Thus the state j l i e s in some domain D(i) in configuration space centered on state i , with volume V(D) = N(2A) . More concisely, 3

q(i,j) = W(D)

j c=D(i)

= 0

otherwise

(3)

Other relevant features of Q are q(i,j)

0

(4)

E q(i,J) = 1 j

(5)

q(i.j) = q ( j » i )

(6)

The t r i a l step generated by £ is accepted with probability min[l ,TT(j)/TT(i)]. A variation on this scheme, usually attributed to A. A. Barker ( 7 J , is to use a step acceptance probability of 1/[1 + Tr(i)/7r(j)]. It has been noted (3) that this algorithm

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

14. OWICKI

Optimization of Sampling Algorithms

161

actually was used earlier by Flinn and McManus (8). To compute ensemble average properties (e.g., the mean potential energy) the relevant mechanical variable is formally represented by a vector f, where f ( i ) is the value of the variable in state i . A Markov chain of M states {X(m)} , m=l,2,...M, is then generated using P_, and the average of f is estimated to be f = 1 £f[X(m)] m=l

(7)

The Hastings Formalism. Hastings {9) has published a generalization of the Metropolis algorithm. The essential features are that Q no longer need be symmetric as in eq. (6), and that any of a whole class of matrix functions a may be used to determine step acceptance probabilities. More s p e c i f i c a l l y , p(i,j (8)

= 1 -

Z

p(i,k)

i = j

We will not describe the allowed functional forms for a except to note that the Metropolis and Barker algorithms, generalized for nonsymmetric Q, represent two special cases: ( ^ ( i j ) = min[l,t(j,i)] a (i,j) B

= 1/[1 + t ( i , j ) ]

(9) (10)

where t(i,j)

= [Tr(i)q(i,j)/(Tr(j)q(j,i))]

(11)

Hastings (and Metropolis et a K ) considered only £ which were reversible, i . e . , Tr(i)p(i,j) = 7r(j)p(j,i)

(12)

There exist non-reversible £ which s t i l l satisfy eq. (1), but reversible P_'s generally are simpler to construct and deal with mathematically. The Metropolis and (especially) the Hastings formalisms in principle give the MC practitioner wide latitude in choosing £ for a particular problem. We now turn to the question of computational efficiency. In other words, how should a and £ be chosen to produce a low error ( i . e . , variance) in f for a given investment in computer resources? In addition to empirical and intuitive rules, there now is some useful theoretical guidance on this question.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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COMPUTER MODELING OF MATTER

Optimization of Step Acceptance Matrix a. It has been widely conjectured that the Metropolis a is superior to the Barker a, since 0^(1',j) ^ a ( i , j ) when i f j i f the same £ is B

used for both. The Metropolis scheme will reject fewer t r i a l steps; i t will sample more states and presumably lead to smaller s t a t i s t i c a l errors. Peskun (]0) has proved that this is so, and he has also obtained some much stronger results. In this mathematical work, optimization refers to minimizing the asymptotic variance of f, using to sample from TT, where the asymptotic variance is defined as v(f.T!>P) =

l

i

m

M-*

{ M

C 5 f(X(m))/M]} m=l

v a r

30

(13)

The basic theorem is as follows. For two choices of JP.-J and Pg, assume that £ Pg in the sense that p-|(i,j) ^ P ( » J ) 1

for a l l ( i , j ) where i + states are at least l i k e l y for

as for Pg.

Then

vd.TL.P^ 4 v(f,7r,P ). 2

It is clear that P^Pg

d> hence, that P^ gives an asymptotic

an

variance which is at best as low as that for P^.

For a discussion

of the special case of sampling in a two-state system, see (11). Peskun also showed that P^P. for any other choice of a within the Hastings formalism, so that the Metropolis scheme is asympt o t i c a l l y the optimum choice of a. Optimization of Transition Matrix Q. In another paper (12), Peskun has investigated how the choice of Q affects the asymptotic variance, again within the Hastings formalism. The optimum (£, which may not be unique (V3), is asymmetric and reversible: ir(1)q(1,j) = 7r(j)q(j,i)

(14)

If £ is reversible, then 0^(1",j) = 1 for a l l ( i , j ) , and P = Q. In other words, i t is asymptotically optimum to place the whole burden of the sampling on Q_ so that no t r i a l steps are rejected. Eq. (14) does not give sufficient constraints to f i x Peskun has shown further that the £ which minimizes v(f,TT,P) is a complicated function both of f and TT. Even though a rigorous optimization is impractical, by making a simplifying approximation he was able to come up with qualitative guidelines for choosing £ to reduce v(f,7r,P). The prescription is to relate [ q ( i . j ) - ir(j)] roughly linearly to [f(D

" f(j)]

2

Tl(i)

TT(i) >, Tl(j)

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(15)

14. OWICKI or

Optimization of Sampling Algorithms [f(1) - f ( j ) ]

2

ir(j)2/7r(1)

163 Tr(i) < ir(j)

At the same time, £ should approximately satisfy eq. (14) and must rigorously satisfy eq. (4) and eq. (5). The prescription implies that transitions should be encouraged to states of high TT and to states with values of f which d i f f e r greatly from that of f in the current state. Next we will discuss how the usual choice of Q can be analyzed in terms of the guidelines. If Q is chosen according to eq. (3), f is ignored completely. The current state i of the system probably will have high TT, as will nearby states j . "Nearby" means states with similar molecular coordinates. We wish to sample these states more frequently than distant states, about which we have no information except that most of them have TT~0 (in a dense f l u i d ) . A simple procedure is to set q ( i » j ) = 0 except for states lying in D(i), where D(i) corresponds to the perturbation of the coordinate i. V(D) should be made including a high proportion of states with high TT. In practice, V(D) is adjusted empirically by increasing A until the step acceptance fraction drops to - 0 . 5 . Thus, there is some optimization of with respect to TT. Within D(i), q ( i , j ) should be made reversible. Unfortunately, this requires extensive knowledge of the behavior of TT in D(i), which is computationally expensive to obtain. A zero-order approximation is that TT is constant in D(i); this leads to the usual symmetric choice q ( i , j ) = q ( j , i ) « It often is not too expensive to obtain spatial analytical derivatives of potential energy functions, as long as the energy is being calculated anyway. One can make the first-order approximation that the gradient of the energy is constant in D(i) and is equal to i t s value at state i , then construct q ( i , j ) to be reversible under this assumption. Pangali et a l - (14 and in this Symposium) have used essentially this technique to improve the rate of convergence of MC calculations on liquid water. A related method was used earl i e r for a quantum f l u i d , but with less success (1_5). The extension to higher derivatives of the energy is obvious. Another approach is to improve by using information about the current configuration to select the size, shape, and position of D(i) at each step. For example, Rossky e l a l . (1_6) used the f i r s t derivative of the energy to select the origin of a Gaussian perturbation of the molecular coordinates. In optimizing Q for a dense f l u i d , the importance of f usually will be less than that of TT. The primary reason is that TT is a strongly varying exponential function of the molecular coordinates, differing significantly from zero only in a small fraction of the states. The variation of most other functions (e.g., the potential energy) is weak by comparison. Since the true ensemble average of f is the sum of f(i)ir(i) over a l l states, states with high TT dominate ?. Also, MC runs typically involve

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COMPUTER

MODELING O F M A T T E R

the calculation of several properties of the system; one £ genera l l y will not optimize a l l of the estimates. Nevertheless, optimizing Q_ with respect to ? sometimes can be important. For example, the Helmholtz free energy difference A(1)-A(0) between two systems 1 and 0 can be written as A(l) - A(0) = -kT-ln<exp[-(U(l)-U(0))/kT]>

(16)

0

where U(l) and U(0) are the potential energies and <> is the ensemble average in the 0 system. Here f, which is the exponent i a l of the energy difference, may be as strongly varying as TT, where TT<* exp[-U(0)/kT]. Configurations where f is large but TT is small may dominate ?. The umbrella sampling technique {Z 1_7) seeks to alter the distribution sampled from, in order to give more equal weight to states with large f but small TT and those witf small f but large TT. Though the approach is different from that of Peskun, similarities do exist in the ways in which f_ enters into the construction o f is also important in the optimization of Q when f is constant over large classes of states. An example is in dilute solutions, where the most important solution properties are determined largely by solute-solvent and solvent-solvent interactions within the immediate solvation shell of the solute, and are relatively insensitive to the configurations of distant solvent molecules. Owicki and Scheraga ( 5 ) proposed an algorithm in which solvent molecules near the solute are perturbed more often than are distant ones. This was applied to calculations of the properties of dilute aqueous solutions of methane (]S) and to calculations of the solvation free energy of hard spheres in the Lennard-Jones f l u i d ( 1 9 ) . Part II of this chapter discusses a generalization of this algorithm. Squire and Hoover (20) used a similar but simpler procedure to calculate the free energy of vacancy formation in a Lennard-Jones crystal. In terms of the Peskun prescription, sampling preferentially near the solute makes q ( i , j ) large for transitions which are likely to involve large values of [ f ( i ) - f ( j ) ] , where f is a property sensitive to intermolecular interactions within the solvation shell. 0

9

2

How Useful are Peskun's Results? One obvious practical drawback is that a sampling algorithm with low asymptotic variance but high computational cost per step may be less efficient overall than another algorithm with higher asymptotic variance but lower cost per step. In fact, conflict of this type may well be common. Another drawback is that optimizing the asymptotic variance does not necessarily optimize the variance in an actual calculation, which consists of a f i n i t e number of steps. Whether the typical length of ~10 steps produces variances which behave asymptotically depends on factors such as the presence of quasi-ergodic problems (11). 6

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14.

OWICKI

Optimization of Sampling Algorithms

165

Nevertheless, Peskun's results seem to be in gratifying agreement with intuition and with empirical observations. Prescription (12) is the best (and only?) mathematical guidance yet published for the optimization of even though i t is hard to apply quantitatively. It is particularly important because i t makes an estimate of the relative significance of f_ and TT in the optimization. Clearly, this f i e l d is f e r t i l e ground for further innovations by mathematicians and MC practitioners alike. II.

An Algorithm for Sampling Preferentially Near the Solute

As was mentioned above, i f one is interested in the structure and energetics of the solvation sphere about one solute molecule dissolved in N solvent molecules, i t should be efficient to concentrate the sampling predominantly on those solvent molecules which are in the solvation sphere. A general algorithm based on this idea is outline Description of the Algorithm. Assume that a weighting function a)(r) has been defined, where r is the distance between a solvent molecule and the solute. The function co may be chosen fairly arbitrarily. It need not be continuous, but i t should be fast to evaluate, nonnegative, and a decreasing function of r. If the system is in state i , the next molecule to be perturbed is selected from the probability distribution W(i)=(W (i),W (i), 1

2

•••W (i)), where the components are defined by N

w (i) =

(17)

K

and the r^ are the solute-solvent distances in state i .

Obviously,

W(i) is normalized. Once selected, the molecule is perturbed in the usual fashion. The sampling from W(i) can be implemented in several ways. For example, a rejection sampling algorithm is outlined in Figure 1. An alternative is direct sampling: a random number £ is chosen uniformly on [0,1], and molecule K is selected, where K-l K Z W. (i) < E, 4 I L=0 L=0 L

and Wg(i) = 0.

W. (i)

(18)

L

Which algorithm is more e f f i c i e n t depends on N

and the degree to which W(i) differs from a uniform distribution. Selecting a molecule from W(i) makes ^ asymmetric, because in general W^(i) f ^ ( j ) . The proper Metropolis step acceptance probability is

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

166

COMPUTER MODELING OF MATTER

J

.

SYSTEM IS IN STATE I

RANDOMLY RANDOMLY PICK PICK MOLECULE £ C [o, l]K

PERTURB K, GENERATING STATE j

CALCULATE

W^j)

1

ACCEPT STATE j WITH PROBABILITY «

Figure 1. Rejection sampling algorithm for sampling from W(i). V/ (i) is the largest element of W(i). irAX

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

14. OWICKI

Optimization of Sampling Algorithms

167

min[l, W (j) Tr(j)/(w (i)7T(i))] K

(19)

K

Example: Computation of the Solute-Solvent Radial D i s t r i bution Function g(r). The following calculation was done to test the effectiveness of a preferential sampling algorithm for reducing the statistical errors in the computed values of g(r) in a (two dimensional) hard disc solution for small values of r, where g(r) differs significantly from unity. The solution was constructed from N = 48 solvent particles with diameter a and one solute particle with diameter 1.5cr. Square periodic boundary conditions were used, with cell side length 8.617a. This corresponds approximately to 0.57 times the close-packing density of the solvent. With the maximum coordinate perturbation A = 0.2a, about 50% of the t r i a l steps were accepted. The solute was perturbed every tenth step. An i n i t i a l regular configuration was "equilibrated" to serv four computations of g(r) algorithms. The computed g(r), averaged over the four runs, is given in Figure 2. The length of each run was 300,000 steps. In the f i r s t run, the solvent particles were sampled uniformly; in the second, a weighting function co(r) = 1/r and the rejection technique in Figure 1 were used to sample preferentially near the solute. Figure 3 shows the resulting distribution W for a typical configuration. The calculated standard errors (21J of the calculated values of g(r) are given in Figure 4. For r^2.3a, the preferential sampling results (denoted by +) are more precise than those obtained with uniform sampling of the solvent particles (Q). In this region, the average of the ratios of the errors is -0.8. If the errors are behaving asymptotically, equal errors will obtain i f roughly (0.8) ~0.6 times as many steps are made in the preferential sampling run as in the uniform sampling run. Although there is a lot of scatter in the error data, we feel that this is a reasonable estimate of the efficiency i n crease. For r^2.3a, there appears to be no clear-cut difference between the two sets of errors. One would expect uniform sampling to give somewhat higher precision above r~2Jo where, our data show, W typically drops below 1/N. Preferential sampling decreased the fraction of steps accepted only slightly, from 0.50 to 0.48. In most molecular systems, the added costs of implementing preferential sampling are a small fraction of the total. If MAX ^ b l e - a v e r a g e largest value of W in a configuration, the mean number of t r i a l s to select a molecule by the rejection sampling algorithm in Figure 1 is • N. In the 2

2

W

1

S

e

ensem

calculation reported here, W n N was found to be - 5 . M

Y

#

The extra

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

168

COMPUTER MODELING OF MATTER

3-5 3-0 2-5 2-0 CO

^

1-5 1-0 0.5

0-0 Figure 2. Computed solute-solvent radial distribution function g(r) in hard disc system. Unit distance is solvent diameter
PARTICLE NUMBER

Figure 3. Probability distribution function W for a typical hard disc configuration, with o>(r) = 1/T . W., .V is the largest element, and the dashed line near W = 0.02 represents a uniform distribution, W = 2 / N for each particle. 2

M

L

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

14. OWICKI

Optimization of Sampling Algorithms

9 3

+

•°+ am

+

o •

a

++

+

1-0

1-5

2-0

169

2-5

. +°

3-0

•

+

+ +

^ •*>

+ &•-•

3-5

+

4-0

4-5

Figure 4. Comparison of standard errors for g(r), when g(r) is computed with different sampling algorithms. (Q) uniform sampling of solvent particles, solute perturbed every tenth step; (-\-) preferential sampling of solvent particles, solute perturbed as above; (#) uniform sampling of solvent particles, solute fixed. For clarity, only every other data point is plotted. Estimates of errors are based on fluctuations in g(r) between blocks of 20,000 configurations (21).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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COMPUTER MODELING OF MATTER

computing costs, compared with uniform sampling, were about 20 seconds of IBM 370/168 CPU time per 10 steps. 5

Digression: How Important Is It to Move the Solute? Perturbing a solvent molecule generates only one new solute-solvent distance for calculating g(r), while moving the solute generates N new distances. This suggests that one should move the solute more often than a solvent molecule, on the average. Note that i t is not necessary to move the solute at a l l , since only relative molecular coordinates are important and thus the coordinate system can be fixed on a molecule ( e . g . , the solute). To test the importance of this effect, the two computations discussed above were repeated, except that the solute was not perturbed. The standard errors for the uniform sampling run are plotted (#) in Figure 2. They usually l i e well above the errors from both runs in which the solute moved For c l a r i t y the solute-fixed/ preferential sampling result different from the solute-moved/preferentia small r , but are significantly higher for large r. Not surprisi n g , i t is important to perturb the solute. Acknowledgements Thanks go to P. Peskun for a copy of his paper {]2) as well as for interesting discussions of the relevance of his results to calculations on dense fluids. The author is supported by an Institutional Research Fellowship under NIH Grant No. GM 07026. Literature Cited 1. 2.

3. 4. 5. 6. 7. 8. 9. 10. 11.

Hammersley, J . M . , and Hanscomb, D. D., "Monte Carlo Methods," especially Chapter 9, Methuen, London, 1964. Valleau, J. P., and Torrie, G. M . , Chapter 5 in Berne, B. J. (ed.), "Statistical Mechanics, Part A," Plenum, New York, 1977. Wood, W. W., and Erpenbeck, J. J., Ann. Rev. Phys. Chem., (1976) 27, 319. Bennett, C. H., J. Comp. Phys., (1976), 22, 245. Owicki, J . C . , and Scheraga, H. A., Chem. Phys. L e t t . , (1977), 47, 600. Metropolis, N . , Rosenbluth, A. W., Rosenbluth, M. N . , T e l l e r , A . H . , and T e l l e r , E., J. Chem. Phys., (1953), 21, 1087. Barker, A. A . , Austral. J. Phys., (1965), 18, 119. Flinn, P. A . , and McManus, G. M . , Phys. Rev., (1961), 124, 54. Hastings, W. K . , Biometrika, (1970), 57, 1. Peskun, P. H . , Biometrika, (1973), 60, 3. Valleau, J . P . , and Whittington, S. G . , Chapter 4 in Berne, B. J . (ed.), "Statistical Mechanics, Part A," Plenun, New York, 1977.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

14. OWICKI Optimization of Sampling Algorithms 12.

13. 14. 15. 16. 17. 18. 19. 20. 21.

171

Peskun, P. H . , "Guidelines for Choosing the Transition Matrix in Monte Carlo Sampling Methods Using Markov Chains," presented at the Fourth Conference on Stochastic Processes and Their Applications, York Univ., Toronto, August 5-9, 1974. For an abstract, see Peskun, P. H . , Adv. Appl. Probab., (1975), 7, 261. Peskun, P. H . , private communication. Pangali, C . , Rao, M . , and Berne, B. J., Chem. Phys. L e t t . , in press. Cepereley, D . , Chester, G. V . , and Kalos, M . , Phys. Rev. B, (1977), 16, 3081. Rossky, P. J., Doll, J . D . , and Friedman, H. L., J. Chem. Phys., in press. Torrie, G . , and Valleau, J. P., J. Comp. Phys., (1977), 23, 187. Owicki, J . C . , and Scheraga, H. A., J. Am. Chem. S o c , (1977), 99, 7413. Owicki, J . C . , an 82, 1257. Squire, D. R., and Hoover, W. G., J. Chem. Phys., (1969), 50, 701. Wood, W. W., Chapter 5 in Temperley, H. N. V . , Rowlinson, J . S., and Rushbrooke, G. S. (eds.), "Physics of Simple Liquids," North-Holland, Amsterdam, 1968.

RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

15 Molecular Dynamics Simulations of Simple Fluids with Three-Body Interactions Included J. M. HAILE Chemical Engineering Department, Clemson University, Clemson, SC 29631

A primary goal i n an ability to p r e d i c t all only information about systems of i n t e r e s t here are those i n which the temperature and molecular masses are sufficiently high that quantum e f f e c t s are n e g l i g i b l e and f o r which the classical Hamiltonian can be assumed separable i n t o independent t r a n s l a t i o n a l , r o t a t i o n a l , i n t e r n a l , and c o n f i g u r a t i o n a l c o n t r i b u t i o n s . The p r i n c i p a l remaining diffi­ culty i n r e a l i z i n g the goal of complete property p r e d i c t i o n f o r such systems i s q u a n t i t a t i v e d e s c r i p t i o n of the i n t e r m o l e c u l a r f o r c e s w i t h i n the fluid. These f o r c e s are commonly discussed i n terms of an i n t e r m o l e c u l a r p o t e n t i a l energy U which, f o r a f l u i d of N s p h e r i c a l molecules, depends only on the p o s i t i o n s of the molecular mass centers r : i

— l

U = U(r

r

r , . . . r^)

(1)

2

This i n t e r m o l e c u l a r p o t e n t i a l i s u s u a l l y w r i t t e n as a s e r i e s of terms which i n d i v i d u a l l y account f o r 2-body, 3-body,... N-body interactions: U - 11 i<j

"(1^ ) + H l i<j
u ( r . r v^)

J

i
^

J

^

(2)

J

L

Much progress i n recent years has been made towards understanding the f l u i d s t a t e v i a the " p a i r theory" of f l u i d s wherein the s e r i e s i n eq. (2) i s truncated a f t e r the f i r s t term (see, f o r example, reference ( l ) ) , i . e . , U - T 7 u(r.r.)

(3)

Even the simplest r e a l f l u i d s (the i n e r t gases) do not obey the pairwise a d d i t i v e assumption eq. (3) at moderate to high d e n s i t i e s (2). However, the advent of the Monte Carlo (3) and

0-8412-0463-2/78/47-086-172$05.00/0 ©

1978

American Chemical Society

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

15.

HAILE

MD Simulations of Fluids with Three-Body Interactions

173

molecular dynamics (4) techniques f o r s i m u l a t i n g f l u i d s on com­ puters has enabled generation of a vast amount of data f o r model f l u i d s which obey eq. (3) e x a c t l y . When compared with v a r i o u s t h e o r e t i c a l c a l c u l a t i o n s (such as from i n t e g r a l equations, per­ t u r b a t i o n t h e o r i e s , etc.) the s i m u l a t i o n r e s u l t s provide t e s t s of the t h e o r i e s which are c o n s i s t e n t i n the assumption of p a i r w i s e additivity. When compared with experimental measurements of the p r o p e r t i e s of r e a l f l u i d s , the s i m u l a t i o n r e s u l t s can be used to d i r e c t l y t e s t the adequacy of the p a i r w i s e a d d i t i v e assumption. The r e s u l t s obtained from these kinds of s t u d i e s have enhanced our understanding of the r e l a t i o n s between features of the i n t e r molecular p o t e n t i a l and f l u i d p r o p e r t i e s . This new knowledge has, i n turn, prompted refinements i n e x i s t i n g p a i r p o t e n t i a l models and c o n s t r u c t i o n of new p o t e n t i a l f u n c t i o n s . Although the computer s i m u l a t i o n of f l u i d s i s now recognized as a powerful t o o l f o r t e s t i n g models f o r the i n t e r m o l e c u l a r p o t e n t i a l , almost no s i m u l a t i o even, three body f o r c e d e t e r r e n t to performing such s i m u l a t i o n s has been the e x c e s s i v e l y l a r g e amount of computer time r e q u i r e d . T h i s paper r e p o r t s devel­ opment of a method f o r performing molecular dynamics s i m u l a t i o n s i n acceptable amounts of computer time f o r 108 s p h e r i c a l molecules with long range, three body i n t e r a c t i o n s e x p l i c i t l y i n c l u d e d . The technique reported here i s a v a r i a n t of the m u l t i p l e time step methods r e c e n t l y developed by S t r e e t t and coworkers (5). A des­ c r i p t i o n of the m u l t i p l e time step method a p p l i e d to n o n s p h e r i c a l molecules i s given elsewhere i n t h i s volume. Of the s e c t i o n s which f o l l o w : S e c t i o n 1 gives a b r i e f summary of previous s t u d i e s of n o n a d d i t i v i t y e f f e c t s : S e c t i o n 2 contains a d e s c r i p t i o n of the t r i p l e d i p o l e A x i l r o d - T e l l e r p o t e n t i a l used i n t h i s study; S e c t i o n 3 presents the molecular dynamics method developed; S e c t i o n 4 r e p o r t s p r e l i m i n a r y r e s u l t s f o r the three body e f f e c t s on e q u i l i b r i u m p r o p e r t i e s . 1. Summary of Previous

Work on

Nonadditivity.

Over the years a number of attempts have been made to c a l c u ­ l a t e nonadditive c o n t r i b u t i o n s to f l u i d and c r y s t a l l i n e proper­ ties. Most of t h i s work has been concerned with nonadditive three body c o n t r i b u t i o n s to e q u i l i b r i u m p r o p e r t i e s , p a r t i c u l a r l y : v i r i a l c o e f f i c i e n t s (6, l_ 8^ j), 10), i n t e r n a l energy (11), f r e e energy (12), and the r a d i a l d i s t r i b u t i o n f u n c t i o n (10, _13, 14>15). A d d i t i o n a l references are c i t e d by Barker, et_ a l . (11). In c o n t r a s t to the l a r g e number of t h e o r e t i c a l s t u d i e s , extremely few computer s i m u l a t i o n s t u d i e s of n o n a d d i t i v i t y e f f e c t s have been performed. T h i s dearth of s i m u l a t i o n work i s due to the p r o h i b i t i v e l y l a r g e amount of c e n t r a l processor time r e q u i r e d to sample N(N-l)(N-2)/3! t r i p l e t s i n a system of N par­ ticles. Barker and coworkers (11) have used both Monte C a r l o and molecular dynamics to estimate three body c o n t r i b u t i o n s t o y

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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MODELING OF MATTER

the i n t e r n a l energy and pressure of l i q u i d argon. However, the nonadditive c o n t r i b u t i o n s to the p r o p e r t i e s were obtained by a p e r t u r b a t i v e technique i n which averages o v e r the three body po­ t e n t i a l were evaluated from a subset of the e n t i r e s i m u l a t i o n (every 1000th c o n f i g u r a t i o n i n the case of Monte C a r l o ) . Re­ peated, time-consuming sampling of a l l p o s s i b l e t r i p l e t s i n the f l u i d was thereby avoided. Recently, Schommers has performed molecular dynamics s i m u l a t i o n s of a two-dimensional Lennard-Jones f l u i d ; the c a l c u l a t i o n s were done i n two dimensions r a t h e r than three because there are s i g n i f i c a n t l y fewer t r i p l e t s of molecules i n two dimensions (16). As study of nonadditive c o n t r i b u t i o n s to p r o p e r t i e s has de­ veloped, u n c e r t a i n t y has a r i s e n as to the r e l a t i v e importance of short and long range three body i n t e r a c t i o n s . Long range i n t e r ­ a c t i o n s are u s u a l l y taken to be i n the form of a m u l t i p o l e expan­ s i o n of d i s p e r s i o n f o r c e s (17) i n which the A x i l r o d - T e l l e r t r i p l e d i p o l e p o t e n t i a l (18, 19 i n t e r a c t i o n s are o f t e n charge overlap or exchange f o r c e s . In the s i m u l a t i o n work by Barker, et_ a l . the A x i l r o d - T e l l e r p o t e n t i a l was assumed to be the only s i g n i f i c a n t nonadditive i n t e r a c t i o n (11, 12). The simu­ l a t i o n using the A x i l r o d - T e l l e r model with a true p a i r p o t e n t i a l gave values of pressure and i n t e r n a l energy i n good agreement with experimental values f o r l i q u i d argon. On the other hand, Jansen and Lombardi argue that short range three body overlap i n t e r a c t i o n s e x p l a i n the s t a b i l i z a t i o n of r a r e gas c r y s t a l s i n an fee s t r u c t u r e r a t h e r than the expected hep s t r u c t u r e (20). T h i s question of the r e l a t i v e importance of short and long range nonadditive i n t e r a c t i o n s apparently remains unresolved. The magnitudes of three body c o n t r i b u t i o n s to f l u i d proper­ t i e s have been found to depend on the s e n s i t i v i t y of the property to the i n t e r m o l e c u l a r p o t e n t i a l . Thus, Barker, et a l . found that the A x i l r o d - T e l l e r p o t e n t i a l c o n t r i b u t e s about 50% to the p r e s ­ sure but only about 5% to the i n t e r n a l energy i n l i q u i d argon (12). The e f f e c t of three body f o r c e s on the r a d i a l d i s t r i b u t i o n f u n c t i o n i s g e n e r a l l y considered to be small (10); although Singh, e_t a l . c l a i m s i g n i f i c a n t three body e f f e c t s on the r a d i a l d i s t r i b u t i o n f u n c t i o n based on thermodynamic p e r t u r b a t i o n theory c a l c u l a t i o n s f o r a Lennard-Jones plus A x i l r o d - T e l l e r f l u i d (14). Barker has obtained the r a d i a l d i s t r i b u t i o n f u n c t i o n f o r l i q u i d argon from a s i m u l a t i o n using the Barker-Fisher-Watts (BFW) p a i r p o t e n t i a l plus the A x i l r o d - T e l l e r model and quantum c o r r e c t i o n s (13, 21). T h i s r a d i a l d i s t r i b u t i o n f u n c t i o n agrees very w e l l with that obtained from neutron d i f f r a c t i o n experiments by Y a r n e l l , et_ a l . (13). However, the Lennard-Jones r a d i a l d i s t r i ­ b u t i o n f u n c t i o n determined by V e r l e t (22) using molecular dyna­ mics agrees e q u a l l y w e l l with the neutron s c a t t e r i n g r e s u l t s (13, 21). The c o n c l u s i o n would seem to be that the r a d i a l d i s ­ t r i b u t i o n f u n c t i o n i s not very s e n s i t i v e to the d e t a i l s of the p o t e n t i a l and, e s p e c i a l l y at high d e n s i t y , i s l a r g e l y determined

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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by short range, r e p u l s i v e f o r c e s . Singh, et_ a l . argue, however, that the BFW and Lennard-Jones are very d i f f e r e n t p a i r p o t e n t i a l s and, t h e r e f o r e , t h e i r r a d i a l d i s t r i b u t i o n f u n c t i o n s must be very different. Hence, they expect the three body c o n t r i b u t i o n to the r a d i a l d i s t r i b u t i o n f u n c t i o n to be l a r g e (14). Schommers twodimensional Lennard-Jones plus A x i l r o d - T e l l e r s i m u l a t i o n s show small e f f e c t s of the three body i n t e r a c t i o n on the r a d i a l d i s t r i ­ b u t i o n f u n c t i o n (16). L i t t l e i s known concerning three body c o n t r i b u t i o n s to dyna­ mic f l u i d p r o p e r t i e s . F i s h e r and Watts have c a l c u l a t e d the s e l f d i f f u s i o n c o e f f i c i e n t using the BFW p a i r p o t e n t i a l without i n c l u ­ ding the A x i l r o d - T e l l e r i n t e r a c t i o n and obtained r e s u l t s s i m i l a r to those f o r a Lennard-Jones f l u i d at the same d e n s i t i e s (23). Schommers two dimensional Lennard-Jones plus A x i l r o d T e l l e r s i m u l a t i o n s show s i g n i f i c a n t three body e f f e c t s on the v e l o c i t y autocorrelation f u n c t i o n ; however, the two dimensional s e l f diffusion coefficient i f u l to point out, however f l u i d s do not n e c e s s a r i l y e x t r a p o l a t e to three dimensions. L i t t l e study has been made of nonadditive e f f e c t s i n f l u i d s composed of n o n s p h e r i c a l molecules. Singh and coworkers have developed a thermodynamic p e r t u r b a t i o n theory f o r t h i s purpose (24, 25). Stogryn has given the analog of the A x i l r o d - T e l l e r p o t e n t i a l f o r n o n s p h e r i c a l molecules (26). 1

1

2.

P o t e n t i a l Model Used

The p o t e n t i a l model used to t e s t the molecular dynamics method described below was a Lennard-Jones 6, 12 p a i r p o t e n t i a l plus A x i l r o d - T e l l e r t r i p l e t p o t e n t i a l . The Lennard-Jones model i s given by:

w h i l e the A x i l r o d - T e l l e r model i s given by u

(17,

18,

19):

, v vTl + 3cos9 cos0 cos8 ] ( r . . , r . , ,r., ) _ 1 2 3 AT ij ik jk (r. . r r ) i j ik jk' n

o

o

,c\ (5)

J

A T

where r \ . i s the s c a l a r d i s t a n c e between atoms i and j , e t c . , 6^ i s the angle formed at atom i by r ^ and r±\ e t c . , and v i s a species dependent constant. The value used f o r v i n t h i s 2 § £ i s thatj quoted by Barker, et a l . (12) f o r argon: v =73.2(10 ) erg cm . Using the u s u a l values f o r the Lennard-Jones p a i r potent i a l p a r a m e t e r s , namely e/k= 120. °K, a= 3.405 A, gives v* = v/ecT = 0.0718. Figures 1 and 2 show the A x i l r o d - T e l l e r p o t e n t i a l f o r sev­ e r a l r e p r e s e n t a t i v e t r i a n g u l a r shapes formed by three atoms. These f i g u r e s i n d i c a t e that the A x i l r o d - T e l l e r p o t e n t i a l i s i9

w

q

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COMPUTER MODELING OF MATTER

Figure 1. Axilrod-Teller triple dipole potential for representative triangles formed by three atoms. The angle 6j is that between r and r . J2

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Figure 2. Axilrod-Teller triple dipole potential for representative triangles formed by three atoms. The angle B is same as in Figure 1. t

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178

COMPUTER MODELING OF MATTER

r e p u l s i v e over most acute t r i a n g l e s and a t t r a c t i v e over most ob­ tuse t r i a n g l e s . The r e p u l s i v e p a r t s of the i n t e r a c t i o n appear to be stronger and of longer range than the a t t r a c t i v e p o r t i o n s . In the simulations reported here the Lennard-Jones p o t e n t i a l was truncated at v = 2.5a. Figures 1 and 2 i n d i c a t e that the A x i l r o d - T e l l e r p o t e n t i a l i s of s h o r t e r range than the LennardJones model f o r most t r i a n g u l a r shapes; hence, the A x i l r o d - T e l l e r p o t e n t i a l was truncated when any one of the lengths *^-9 i^> r.. was greater than r = 2.5a: jk c Q

r

o

r

1 J

&

u

AT

( . . ,r.. ,r.. ) = 0 IJ ik' jk

i f r . . , r , or r , > r i j ' lk'

r

c

A s i m i l a r t r u n c a t i o n of t h i s p o t e n t i a l was (12). 3.

(6)

used by Barker, et a l .

Molecular Dynamics with Three-Body I n t e r a c t i o n s

3.1 Conventional Molecula i n t e r a c t i o n s are e x p l i c i t l y i n c l u d e d i n molecular dynamics simu­ l a t i o n s , the equations of motion to be solved take the form: 2

d r. at F, = F . — i —

2 B

2B F. =2 B

F. _

1

3B

+ F. l — i r I i*j

9

v v = - H j<^i

3 B

u

2

(8)

B

(

r

ii> . *J ^ i 3 u

( r

(9)

r

i1» ik , ^± 3

l

k

, r

lk "

)

lk

(10)

3

where _F., n^, r_^ are the f o r c e , mass and p o s i t i o n v e c t o r of mass center r o r p a r t i c l e i , r e s p e c t i v e l y , and s u p e r s c r i p t s 2B and 3B denote two and three body c o n t r i b u t i o n s , r e s p e c t i v e l y . In the work reported here these equations of motion were solved using a p r e d i c t o r - c o r r e c t o r scheme due to Gear (27). D e t a i l s of a p p l i ­ c a t i o n of t h i s method to molecular dynamics s i m u l a t i o n s are given elsewhere (28, 29). The geometry of the system was taken to be a cube of s i d e L and p e r i o d i c boundary c o n d i t i o n s were used with the u s u a l minimum image c r i t e r i o n (30). In molecular dynamics with only p a i r i n t e r a c t i o n s , the speed of program execution i s l i m i t e d by e v a l u a t i o n of the f o r c e on each p a r t i c l e , eq. (9). Since simulations u s u a l l y u t i l i z e p a i r p o t e n t i a l s truncated at some p a i r s e p a r a t i o n r
In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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m a i n t a i n i n g some type o f l i s t of p a r t i c l e s i n order to avoid sampling those p a r t i c l e s separated by d i s t a n c e s g r e a t e r than r ^ . At l e a s t two types of such l i s t s are i n use. One i s a neighbor l i s t (31) maintained f o r each p a r t i c l e i which t a b u l a t e s a l l p a r t i c l e s that a r e w i t h i n a d i s t a n c e r of the p a r t i c l e i . The l i s t d i s t a n c e r i s chosen to be s l i g h t l y longer than the c u t o f f d i s t a n c e r . The second type i s a c e l l l i s t (32, 33) i n which the system i s d i v i d e d i n t o s m a l l c e l l s and a l i s t i s maintained of the p a r t i c l e s w i t h i n each c e l l . In t h i s method, p a i r i n t e r ­ a c t i o n s are sampled only between p a r t i c l e s i n adjacent c e l l s . I n both methods the contents of the l i s t s must be updated a t p e r i o ­ d i c i n t e r v a l s throughout the s i m u l a t i o n . When three body i n t e r a c t i o n s are included i n the s i m u l a t i o n , the speed l i m i t i n g step becomes e v a l u a t i o n of the three body f o r c e s of eq. (10). Now, i f the t r u n c a t i o n of the three body p o t e n t i a l i s taken to be that given by eq. (6) with r having the same v a l u e f o r bot the same l i s t used to spee be a p p l i e d to the double sum i n eq. (10); i . e . , no i n c r e a s e i n storage i s r e q u i r e d . For example, i f a neighbor l i s t i s used, then the double sum i n eq. (10) i s formed only over those mole­ c u l e s h e l d i n the l i s t . Forming t r i p l e t s u s i n g molecules from the l i s t w i t h any molecule not i n the l i s t a u t o m a t i c a l l y g i v e s : r..,r , or r.. > r > r i j * ik' jk L c # 1

T

(11)

and the t r u n c a t i o n o f eq. (6) a p p l i e s . T h i s a l g o r i t h m i n which a neighbor l i s t i s used f o r both two and three body f o r c e e v a l u ­ a t i o n and i n which three body f o r c e s are c a l c u l a t e d e x p l i c i t l y at each time step s h a l l be r e f e r r e d to as the c o n v e n t i o n a l mole­ c u l a r dynamics (CMD) method. U n f o r t u n a t e l y , even with the neighbor l i s t a p p l i e d to the three body f o r c e s , the CMD program f o r 108 Lennard-Jones plus A x i l r o d - T e l l e r p a r t i c l e s executes about 15 times slower than the same program f o r 108 Lennard-Jones p a r t i c l e s . In the work r e ­ ported here, source programs were w r i t t e n i n FORTRAN IV u s i n g s i n g l e p r e c i s i o n a r i t h m e t i c , compiled on an IBM FORTRAN IV-G compiler, and executed on the IBM 370/165 a t Clemson U n i v e r s i t y . 3.2 M u l t i p l e Time Step (MTS) Method A p p l i e d to Three Body Forces. In order to improve the execution speed of simula­ tion programs with three body i n t e r a c t i o n s i n c l u d e d , the mul­ t i p l e time step method of S t r e e t t , e t a l . (5) has been a p p l i e d to the e v a l u a t i o n of the three body f o r c e s . The m u l t i p l e time step method attempts to take advantage of the f a c t that molecular motions i n a f l u i d may be reduced to components which operate on very d i f f e r e n t time s c a l e s . More p r e c i s e l y , one can o f t e n i d e n ­ t i f y components o f the f o r c e on a molecule which have r e l a t i v e l y l a r g e d i f f e r e n c e s i n t h e i r r a t e s of change with time. I t i s the q u i c k l y v a r y i n g component of the f o r c e which l i m i t s the s i z e of the time step At which must be used to o b t a i n s t a b l e s o l u t i o n s

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

180

COMPUTER

MODELING OF MATTER

to the d i f f e r e n t i a l equations of motion. In the MTS method these q u i c k l y v a r y i n g components of the f o r c e are e x p l i c i t l y evaluated at each time step i n the u s u a l way; however, the slowly v a r y i n g components are only e x p l i c i t l y evaluated at longer i n t e r v a l s , say, every n time steps. At time steps between e x p l i ­ c i t determination, the slowly v a r y i n g components of the f o r c e are estimated by e x t r a p o l a t i o n from the previous e x p l i c i t e v a l u ­ ation. The e x t r a p o l a t i o n may be l i n e a r (LMTS method) or, i f one i s more ambitious, based on a T a y l o r s e r i e s expansion. The degree to which the MTS method improves execution speed of the program depends on the amount of computation avoided i n e x t r a ­ p o l a t i n g the slowly v a r y i n g f o r c e components r a t h e r than e x p l i ­ c i t l y c a l c u l a t i n g them. I f we consider a p p l y i n g t h i s MTS method to s i m u l a t i o n s with three body i n t e r a c t i o n s , eq. (8) already represents the f o r c e on molecule i d i v i d e d i n t o two p a r t s Barker and Henderson have noted that the three bod v a r y i n g (21). To t e s s i m u l a t i o n f o r 108 p a r t i c l e s i n t e r a c t i n g with Lennard-Jones plus A x i l r o d - T e l l e r p o t e n t i a l s u s i n g the conventional molecular dyna­ mics algorithm. Figure 3 shows a t y p i c a l comparison of a compo­ nent of the two and three body f o r c e s on one p a r t i c l e during a segment of the s i m u l a t i o n . The f i g u r e i n d i c a t e s that the three body component of the f o r c e does indeed have a slower r a t e of change than the two body f o r c e component. Encouraged by F i g u r e 3, the MTS method was then a p p l i e d to the s i m u l a t i o n i n the f o l l o w i n g manner. The e n t i r e three b o d y ^ f o r c e i s considered to be slowly v a r y i n g , t h e r e f o r e a l l of _F. i s e x p l i c i t l y evaluated only at p e r i o d i c i n t e r v a l s i n ^ t h e simu­ lation. We choose to use l i n e a r e x t r a p o l a t i o n of F. for, although S t r e e t t , et a l . f i n d that a t h i r d order T a y l o r s e r i e s e x t r a p o l a t i o n i s more accurate, the a n a l y t i c e v a l u a t i o n s of the time d e r i v a t i v e s o^ the A x i l r o d - T e l l e r f o r c e are h o p e l e s s l y tedious. Thus, F\ i s e x p l i c i t l y evaluated at two s u c c e s s i v e times, t and (t + A t ) , and then l i n e a r l y e x t r a p o l a t e d over the next n time steps. T h i s gives the three body f o r c e on p a r t i c l e i at any intermediate time step ( t + k At) as: fi

Q

Q

F. — i

3 B

(t + k At) = F . o —1

3 B

3 B

( t ) + k [ F . ( t + At) - F . o — i o —x

3 B

( t )] o

(12)

In a s i m u l a t i o n program using a p r e d i c t o r - c o r r e c t o r a l g o ­ rithm, eq. (12) would appear i n the f o r c e e v a l u a t i o n step. The only a d d i t i o n a l storage r e q u i r e d f o r the MTS method over that f o r the CMD method i s ^ s p a c e f o r 6N numbers - 3N l o c a t i o n s f o r the components of _F. ( t ^ ) * 3N l o c a t i o n s f o r the components of the d i f f e r e n c e term i n brackets i n eq. (12). We experimented with v a r i o u s values of the time step At and number of e x t r a p o l a t i o n steps n. A compromise among s t a b l e s o l u t i o n s of the d i f f e r e n t i a l equations, conservation of system energy, and execution speed was obtained using n = 8 and a n c

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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181

At = 2.15 (10 ) s e c . T h i s value of the time step i s about onef i f t h that used by V e r l e t (31) and Rahman (34) i n s i m u l a t i o n s of the p a i r w i s e a d d i t i v e Lennard-Jones f l u i d . Note t h a t , i n the MTS method, any property which i s being c a l c u l a t e d as a running ensemble average a t each time step during the s i m u l a t i o n must a l s o be d i v i d e d i n t o slowly (three body) and q u i c k l y (two body) v a r y i n g c o n t r i b u t i o n s . The MTS method i s a p p l i e d to the slowly v a r y i n g components of the p r o p e r t i e s v i a equations analogous to eq. (12). T h i s comment a p p l i e s , f o r example, to the i n t e r n a l energy and pressure i n the work reported here. Two p a r a l l e l s i m u l a t i o n s f o r 108 p a r t i c l e s i n t e r a c t i n g with Lennard-Jones p l u s A x i l r o d - T e l l e r p o t e n t i a l s have been performed. The f i r s t c a l c u l a t i o n u t i l i z e d the CMD method i n which the f o r c e s were e x p l i c i t l y evaluated a t each time step. In the second run the two body f o r c e s were determined i n the standard way and the LMTS metho body f o r c e s . Both run t i c l e p o s i t i o n s and v e l o c i t i e s and both were continued f o r 1650 time steps. A comparison of the p r o p e r t i e s obtained from the two c a l c u l a t i o n s i s given i n Table I . In a d d i t i o n to the pro­ p e r t i e s l i s t e d i n Table I , r a d i a l d i s t r i b u t i o n f u n c t i o n s , v e l o ­ c i t y , speed, and f o r c e a u t o c o r r e l a t i o n f u n c t i o n s , and atomic mean squared displacements (from which d i f f u s i o n c o e f f i c i e n t s were obtained) were c a l c u l a t e d . For a l l of these p r o p e r t i e s , the LMTS values were w i t h i n 0.1% of the values obtained by the CMD method. Figure 4 shows the per cent d e v i a t i o n i n the i n s t a n ­ taneous t o t a l energy o f the two c a l c u l a t i o n s . Since, i n the LMTS method d e s c r i b e d above, the three body f o r c e s a r e e x p l i c i t l y evaluated only two of every ten time s t e p s , we might expect the LMTS method to be about 5 times as f a s t as the CMD method. The r e s u l t s i n Table I confirm t h i s ; i . e . , u s i n g the LMTS method, a s i m u l a t i o n of 108 Lennard-Jones p l u s A x i l r o d - T e l l e r p a r t i c l e s r e q u i r e s about 20 CPU minutes f o r 2000 time steps on an IBM 370/165. T h i s execution speed remains about 2.5 times slower than a CMD method, p a i r w i s e a d d i t i v e LennardJones s i m u l a t i o n . The r e s u l t s from the LMTS method and the CMD method a r e i n good agreement f o r those p r o p e r t i e s which a r e averaged over the s i m u l a t i o n . However, a f t e r about 1000 time steps, the i n s t a n ­ taneous components of the three body f o r c e s on i n d i v i d u a l par­ t i c l e s i n the LMTS c a l c u l a t i o n begin to d e v i a t e from those i n the CMD method, as i n d i c a t e d i n Table I . These d e v i a t i o n s a r e due to gradual accumulation o f i n a c c u r a c i e s from the l i n e a r e x t r a p o l a t i o n of the three body f o r c e . I t should be emphasized that while these d e v i a t i o n s occur, the t o t a l energy and momentum of the LMTS system remain c o n s e r v a t i v e and agree c l o s e l y with those of the CMD method (see F i g u r e 4 ) . This disagreement i n the three body f o r c e s can be i n t e r p r e t e d as a d r i f t i n g o f the

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

182

Figure 3. Comparison of z-component of two-(2B) and three-body (3B) forces on particle #27 in conventional molecular dynamics simulation of Lennard-Jones plus Axilrod-Teller fluid. At = time step = 2.15 (10' ) sec, a = 0.65, kTV = 1.033. Note that the scale for three-body force is an order of magnitude smaller than that for the two-body force. 15

P

1

3

%

-0.1 L

1 1400

1

1 1600

I

I 1800

I

1 2000

At

Figure 4. Percent deviation in instantaneous total system energy calculated by MTS method from that determined by CMD method for 108 particles interacting with Lennard-Jones plus Axilrod-Teller forces. System parameters same as those in Figure 3. Percent deviation calculated as: (E MD — EUTS)100/E UD' C

C

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Table I Test of the L i n e a r M u l t i p l e Time Step Method A p p l i e d to the A x i l r o d - T e l l e r P o t e n t i a l i n Molecular Dynamics Simulations of Lennard-Jones p l u s A x i l r o d - T e l l e r I n t e r a c t i o n s . CMD Method

Number P a r t i c l e s , N

108

LMTS Method

108

3 0.65

0.65

2.15

2.15

1.0325

1.0326

Number d e n s i t y , pa Time step, At (10"^)se CPU time, mins. Temperature,

kTe ^ -4.356

-4.356

0.104

0.103

-2.795

-2.797

3B 3 -1 a e

-0.363

-0.373

3B 3 -1 a e

0.125

0.120

3B 3 -1 ae

0.0465

0.0519

xa

1.457

1.462

v e c t o r f o r p a r t i c l e 27

ya

4.981

4.987

at time step 1650

za

1.358

1.364

Average Conf. I n t . Energy, U(Ne) Average Pressure, P(pkT) ^ Instantaneous T o t a l Energy E(Ne) at time step 1650 Instantaneous Components of 3-body Force

p

on p a r t i c l e 27 a t time

r

F

x

F y

step 1650

Components of p o s i t i o n

r

F

z

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

P(pkT)

-1

-1

-1

U(Nc)

kTe

Number Time Steps

Time Step ( 1 0 ) sec

1 5

Number P a r t i c l e s

3

= 0.65

-0.11

-4.52

1.036

1200

9.6

864

-0.05

-4.54

1.052

2000

2.15

108

Verlet This ( r e f . (31)) Work

Lennard-Jones

pa

+0.10

-4.36

1.033

1650

2.15

108

MTS Method

L J + AT

pa

-0.20

-5.90

+0.38

-5.64

0.746

1730

2.15

108

MTS Method

LJ + AT

= 0.817

0.746

2000

2.15

108

This Work

Lennard-Jones

P r e l i m i n a r y R e s u l t s Showing E f f e c t of Three Body A x i l r o d - T e l l e r Force on I n t e r n a l Energy and Pressure

Table I I

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LMTS system onto a d i f f e r e n t t r a j e c t o r y i n phase space from that followed by the system evolved by the CMD method. Hence, there i s no s t r i c t l y Newtonian connection between widely separated phase p o i n t s ; but then, there i s no true Newtonian connection between widely separated phase p o i n t s i n the system generated by the CMD method due to round o f f e r r o r s and the truncated poten­ tial. The c o n c l u s i o n i s that the phase space t r a j e c t o r i e s become d i f f e r e n t i n the two c a l c u l a t i o n s , but one i s not n e c e s s a r i l y more wrong (or r i g h t ) than the other. Hoover and Ashurst have discussed t h i s p o i n t i n another context with s i m i l a r c o n c l u s i o n s (35). The r e s t r i c t i o n which t h i s imposes i s that the LMTS method cannot be used to study very l o n g - l i v e d phenomena; such as the long t a i l of the v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n . 4.

Preliminary

Results

for Equilibrium

Properties

The l i n e a r m u l t i p l has been used to simulat Jones {jlus A x i l r o d - T e l l e r p o t e n t i a l s at two s t a t e c o n d i t i o n s : (a) p a =0.65, kTe~ = 1.036, which i s one of the c o n d i t i o n s of the Lennard-Jones f l u i d s t u d i e d by V e r l e t (31) and by Singh, et a l . (14, 15), (b) p a = 0.817, kTe = 0.746, which i s c l o s e to the c o n d i t i o n of the Lennard-Jones f l u i d s t u d i e d by Rahman (34). From these two c a l c u l a t i o n s the c o n f i g u r a t i o n a l i n t e r n a l energy, pressure, and r a d i a l d i s t r i b u t i o n f u n c t i o n have been determined. For comparison, s i m u l a t i o n runs f o r a p u r e l y Lennard-Jones f l u i d have a l s o been performed at s t a t e c o n d i t i o n s c l o s e to the above. E q u i l i b r i u m property r e s u l t s at both s t a t e c o n d i t i o n s f o r the Lennard-Jones and Lennard-Jones plus A x i l r o d - T e l l e r f l u i d s are compared i n Table I I . Long range c o r r e c t i o n s f o r the truncated two body p o t e n t i a l have been added to the i n t e r n a l energy and pressure; however, long range c o r r e c t i o n s f o r the truncated three body p o t e n t i a l have been neglected i n the v a l u e s given i n Table I I . These r e s u l t s confirm the e a r l i e r work by Barker, et a l . i n that the three body e f f e c t i s only a few percent on the i n t e r n a l energy, but i s s i g n i f i c a n t l y l a r g e r on the pressure. Note t h a t , at the s t a t e c o n d i t i o n s s t u d i e d , the net e f f e c t of the A x i l r o d - T e l l e r i n t e r a c t i o n i s r e p u l s i v e , as evidenced by p o s i t i v e c o n t r i b u t i o n s to the energy and pressure. In Figure 5 the r a d i a l d i s t r i b u t i o n f u n c t i o n s f o r the LennardJones and Lennard-Jones plus A x i l r o d - T e l l e r f l u i d s are compared at the high d e n s i t y s t a t e c o n d i t i o n . I t appears t h a t the three body i n t e r a c t i o n s have no e f f e c t except i n the f i r s t peak r e g i o n where the t r i p l e t i n t e r a c t i o n s lower g(r) by about 3%. However, we estimate the s t a t i s t i c a l e r r o r i n g(r) f o r these s i m u l a t i o n s i s about +3%, so i t i s unclear to what extent the observed effect i s real. We note that Schommers has reported s m a l l three body e f f e c t s on the d i s t r i b u t i o n f u n c t i o n f o r the two dimensional f l u i d , as w e l l (16). However, the p e r t u r b a t i o n theory c a l c u l a 3

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

Figure 5. Effect of Axilrod-Teller potential on radial distribution function at pa = 0.817. For Lennard-Jones fluid, kTY = 0.746; for Lennard-Jones plus Axilrod-Teller fluid, kTY = 0.740. 3

1

1

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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MD Simulations of Fluids with Three-Body Interactions

Figure 6. Comparison of trajectories of particle #27 in Lennard-Jones and Lennard-Jones plus Axilrod-Teller fluids at pa = 0.65. Both runs were started from the same point in phase space and trajectories shown are from time steps 5002000 in each simulation. For this calculation the Axilrod-Teller strength constant v was assigned a value of three times that for argon given in Section 2. Note that the circles represent positions of the center of mass of the atom, not the atomic diameter. 3

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t i o n s of Singh and Ram show " s i g n i f i c a n t e f f e c t of the t r i p l e t i n t e r a c t i o n s on g(r) at low temperatures and high d e n s i t i e s (14). The g(r) f o r the low d e n s i t y c a l c u l a t i o n shows e s s e n t i a l l y no e f f e c t due to the three body f o r c e s . Singh and Ram a l s o f i n d l i t t l e e f f e c t at low d e n s i t i e s . Although some dynamic p r o p e r t i e s have been c a l c u l a t e d from these s i m u l a t i o n s , we f i n d , i n g e n e r a l , that 2000 time steps f o r 108 p a r t i c l e s do not provide s u f f i c i e n t data to o b t a i n s t a ­ t i s t i c a l l y meaningful dynamic p r o p e r t i e s . Work i s now i n pro­ gress aimed at i n c r e a s i n g the number of p a r t i c l e s i n the system and the l e n g t h of the run to improve the s t a t i s t i c a l p r e c i s i o n of dynamic property c a l c u l a t i o n s . In an e f f o r t to g a i n f u r t h e r enlightenment as to the e f f e c t s of the A x i l r o d - T e l l e r p o t e n t i a l , we have compared i n d i v i d u a l p a r t i c l e t r a j e c t o r i e s f o r the same p a r t i c l e over the same time segment from two d i f f e r e n t s i m u l a t i o n s . In one s i m u l a t i o n the intermolecular p o t e n t i a the p o t e n t i a l was Lennard-Jone c u l a t i o n s were done at the same d e n s i t y and were s t a r t e d from the same c o n f i g u r a t i o n and p a r t i c l e v e l o c i t i e s . F i g u r e 6 com­ pares the t r a j e c t o r i e s obtained from such a study. In the Lennard-Jones f l u i d the p a r t i c l e shown i n F i g u r e 6 i s e x h i b i t i n g v i b r a t o r y motion due to c o l l i s i o n s with neighboring p a r t i c l e s . In the Lennard-Jones plus A x i l r o d - T e l l e r f l u i d , the same p a r t i c l e during the same time segment i s undergoing an extended t r a j e c t o r y of l a r g e l y d i f f u s i o n a l motion. I t may be that study of three body f o r c e s on dynamic p r o p e r t i e s w i l l r e v e a l f a i r l y strong e f f e c t s on i n d i v i d u a l p a r t i c l e motion but that these e f f e c t s tend to cancel when averaged to o b t a i n bulk dynamic p r o p e r t i e s . Hence, Schommers two dimensional s i m u l a t i o n s show strong three body e f f e c t s on the v e l o c i t y a u t o c o r r e l a t i o n f u n c t i o n but i n s i g n i f i c a n t e f f e c t on the s e l f d i f f u s i o n c o e f f i c i e n t (16). 1

5.

Conclusions

A m u l t i p l e time step method has been s u c c e s s f u l l y developed f o r performing molecular dynamics simulations i n which three body i n t e r a c t i o n s are e x p l i c i t l y i n c l u d e d . The method has been t e s t e d f o r 108 p a r t i c l e s i n t e r a c t i n g with a Lennard-Jones plus A x i l r o d - T e l l e r p o t e n t i a l . When compared with a conventional molecular dynamics program u s i n g the same p o t e n t i a l model, the MTS program executes - 4 . 5 times f a s t e r while g i v i n g e s s e n t i a l l y the same values f o r e q u i l i b r i u m and dynamic p r o p e r t i e s , f o r runs of up to 2000 time steps. At t h i s p o i n t , the MTS method i s of d o u b t f u l r e l i a b i l i t y f o r studying l o n g - l i v e d phenomena. Based on t h i s study, i t seems that the m u l t i p l e time step methods could provide a mechanism f o r extending molecular dynamics simu­ l a t i o n s to a v a r i e t y of phenomena which i n v o l v e components that i n h e r e n t l y operate on d i f f e r e n t time s c a l e s ; f o r example, study of phase t r a n s i t i o n s or long-chain molecules.

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In p r e l i m i n a r y s t u d i e s of three body e f f e c t s w i t h the MTS program, we have confirmed the work of Barker, et_ a l . (11, 12) that the A x i l r o d - T e l l e r c o n t r i b u t i o n i s only a few per cent f o r the i n t e r n a l energy but s i g n i f i c a n t l y l a r g e r f o r the p r e s s u r e . The e f f e c t of the A x i l r o d - T e l l e r i n t e r a c t i o n on the r a d i a l d i s t i r b u t i o n f u n c t i o n i s found to be s m a l l . This r e s u l t i s i n agreement w i t h Schommers' two dimensional s i m u l a t i o n s (16); but disagrees w i t h the t h e o r e t i c a l c a l c u l a t i o n s of Singh, et_ a l . at high d e n s i t y (14, 15). F i n a l l y , we note that the work reported here begins a study of long range, three body i n t e r a c t i o n s but does not i n c l u d e p o s s i b l e short range, three body e f f e c t s . P. A. E g e l s t a f f i s studying such e f f e c t s u s i n g neutron s c a t t e r i n g and Monte C a r l o s i m u l a t i o n (36). Acknowledgments The author i s g r a t e f u f o r v a l u a b l e d i s c u s s i o n s on the m u l t i p l e time step method; H. W. Graben f o r d i s c u s s i o n s on three body f o r c e s ; P. A. E g e l s t a f f and J . A. Barker f o r i n s t r u c t i v e correspondence. T h i s work was supported, i n p a r t , by a grant from the F a c u l t y Research Committee, Clemson U n i v e r s i t y . The Clemson U n i v e r s i t y Computer Center generously provided the computer time used i n t h i s work. Literature Cited 1. Hansen, J . P. and McDonald, I . R., "Theory of Simple L i q u i d s , " Academic P r e s s , New York, 1976. 2. Watts, R. 0. and McGee, I. J., " L i q u i d State Chemical P h y s i c s , " J . Wiley and Sons, New York, 1976. 3. M e t r o p o l i s , M., Rosenbluth, A. W., Rosenbluth, M. N., Teller, A. N., and T e l l e r , E., J. Chem. Phys. (1953) 21, 1087. 4. A l d e r , B. J . and Wainwright, T. E., J. Chem. Phys. (1957) 27, 1208. 5. S t r e e t t , W. B., T i l d e s l e y , D. J., and Saville, G., Molec. Phys. (1978) in p r e s s . 6. Copeland, D. A. and Kestner,N.R.,J.Chem. Phys. (1968) 49, 5214. 7. Casanova, G., D u l l a , R. J., Jonah, D. A., Rowlinson, J . S., and S a v i l l e , G., Molec. Phys. (1970) 18, 589. 8. D u l l a , R. J., Rowlinson, J. S., and Smith, W. R., Molec. Phys. (1971) 21, 299. 9. Sherwood, A. E. and P r a u s n i t z , J .M.,J.Chem. Phys. (1964) 41, 413. 10. Fowler, R. and Graben, H. W., J. Chem.Phys. (1972) 56, 1917. 11. Barker, J . A., F i s h e r , R. A., and Watts, R. O., Molec.Phys. (1971) 21, 657.

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12.

Barker, J . A., Henderson, D., and Smith, W. R., Molec. Phys. (1969) 17, 579. 13. Y a r n e l l , J . L., Katz, M. J . , Wenzel, R. G., and Koenig, S. H., Phys. Rev. A (1973) 7, 2130. 14. Ram, J . and Singh, Y., J. Chem. Phys. (1977) 66, 924. 15. Sinha, S. K., Ram, J . , and Singh, Y., J . Chem. Phys. (1977) 66, 5013. 16. Schommers, W., Phys. Rev. A (1977) 16, 327. 17. B e l l , R. J., J. Phys. B (1970) 3, 751. 18. A x i l r o d , B. M. and Teller, E., J. Chem. Phys.(1943) 11, 299. 19. Muto, Y., Proc. Phys. Math. Soc. Japan (1943) 17, 629. 20. Jansen, L. and Lombardi, E., Faraday D i s c . Chem. Soc. (London) (1965) 40, 78. 21. Barker, J . A. and Henderson, D., Rev. Mod. Phys. (1976) 48, 587. 22. V e r l e t , L., Phys. Rev. (1968) 165, 201. 23. F i s h e r , R. A. and 25, 529. 24. Singh, Y., Molec. Phys. (1975) 29, 155. 25. Shukla, K. P., Ram, J . , and Singh, Y., Molec. Phys. (1976) 31, 873. 26. Stogryn, D. E., J . Chem. Phys. (1970) 52, 3671. 27. Gear, C. W., "Numerical Initial Value Problems in Ordinary Differential Equations," Prentice-Hall, Englewood Cliffs, New Jersey, 1971. 28. Cheung, P. S. Y. and Powles, J . G., Molec. Phys. (1974) 30, 921. 29. H a i l e , J . M., "Surface Tension and Computer Simulation of Polyatomic F l u i d s , " Ph.D. D i s s e r t a t i o n , U n i v e r s i t y of F l o r i d a , G a i n e s v i l l e , 1976. 30. Wood, W. W., i n "Physics o f Simple L i q u i d s , " H. N. V. Temperley, J . S. Rowlinson, and G. S. Rushbrooke, eds, North-Holland, Amsterdam, 1968. 31. V e r l e t , L., Phys. Rev. (1967) 159, 98. 32. S c h o f i e l d , P., Comput. Phys. Comm. (1973) 5, 17. 33. Quentrec, B. and Brot, C., J. Comput. Phys. (1973) 13, 430. 34. Rahman, A., Phys. Rev. (1964) 136, 405. 35. Hoover, W. G. and Ashurst, W. T. in " T h e o r e t i c a l Chemistry," vol. 1, H. E y r i n g and D. Henderson, eds., Academic Press, New York, 1975. 36. E g e l s t a f f , P. A., p r i v a t e communication (1978). RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16 Monte Carlo Studies of the Structure of Liquid Water and Dilute Aqueous Solutions DAVID L. BEVERIDGE, MIHALY MEZEI, S. SWAMINATHAN, and S. W. HARRISON Chemistry Department, Hunter College of the City University of New York, 695 Park Avenue, New York, NY 10021 The advent o f t h i r gether with recent advance molecular quantum mechanics and statistical mechanics combine to make the s t r u c t u r e of molecular l i q u i d s a c c e s s i b l e to t h e o r e t i c a l study at new l e v e l s of r i g o r v i a computer s i m u l a t i o n . There is p r e s e n t l y broad based research a c t i v i t y in this area from q u i t e d i v e r s e points of view in physics and chemistry. A s e r i e s of Monte Carlo computer s i m u l a t i o n s t u d i e s of the s t r u c t u r e and p r o p e r t i e s of molecular l i q u i d s and s o l u t i o n s have r e c e n t l y been c a r r i e d out in t h i s Laboratory. The c a l c u l a t i o n s employ the canonical ensemble Monte C a r l o - M e t r o p o l i s method based on a n a l y t i c a l pairwise p o t e n t i a l functions r e p r e s e n t a t i v e of ab initio quantum mechanical c a l c u l a t i o n s o f the i n t e r m o l e c u l a r i n t e r a c t i o n s . A number of thermodynamic p r o p e r t i e s i n c l u d i n g in­ ternal energies and radial d i s t r i b u t i o n functions were determined and are reported h e r e i n . The r e s u l t s are analyzed f o r the s t r u c ­ ture o f the statistical s t a t e o f the systems by means o f q u a s i component d i s t r i b u t i o n f u n c t i o n s f o r c o o r d i n a t i o n number and binding energy. S i g n i f i c a n t molecular s t r u c t u r e s c o n t r i b u t i n g to the statistical s t a t e o f each system are identified and d i s p l a y e d i n stereographic form. This a r t i c l e reviews the main r e s u l t s of our most recent work and deals s p e c i f i c a l l y with l i q u i d water, the d i l u t e aqueous s o l u t i o n of methane, and d i l u t e aqueous s o l u t i o n s o f monatomic cations and anions. The background f o r these s t u d i e s i s surveyed i n Section I , followed by general considerations on the method­ ology and computational parameters. Sections III-V c o l l e c t the i n d i v i d u a l r e s u l t s system by system, followed i n Section VI by a general d i s c u s s i o n and conclusions. 1-4

I.

Background The motional degrees o f freedom a v a i l a b l e to molecules i n l i q u i d s and solutions mandate t h e o r e t i c a l s t u d i e s o f these systems to be problems i n s t a t i s t i c a l mechanics and dynamics. The most fundamental approach to problems i n t h i s area i s t o t r e a t each

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system as simply an assembly of molecules i n t e r a c t i n g under a c o n f i g u r a t i o n a l p o t e n t i a l , and t o attempt to solve the c o r r e ­ sponding many-body problem i n c l a s s i c a l s t a t i s t i c a l mechanics or k i n e t i c theory. When the intermolecular i n t e r a c t i o n s are r e l a ­ t i v e l y strong, as i n most molecular l i q u i d s , general s o l u t i o n s cannot be r e a d i l y developed i n a n a l y t i c a l form. However, recent s t u d i e s show i t i s p o s s i b l e t o approach these problems by numeri­ c a l methods using large s c a l e d i g i t a l computers. In s t a t i s t i c a l mechanics, the system can be t r e a t e d by c o n f i g u r a t i o n a l averaging u s i n g the Monte C a r l o - M e t r o p o l i s method, whereas i n the k i n e t i c approach, molecular dynamics, the i n d i v i d u a l molecular t r a j e c t o ­ r i e s are c a l c u l a t e d by simultaneous s o l u t i o n of the Newton-Euler equations. The Monte C a r l o and molecular dynamic methods are c o l l e c t i v e l y r e f e r r e d t o as "computer s i m u l a t i o n s " . In p r i n c i ­ p l e , the computer s i m u l a t i o n methods are able t o accommodate a l l the thermodynamic and r e l a t e d p r o p e r t i e s of an e q u i l i b r i u m s y s ­ tem, and molecular dynamic systems as w e l l . The s i m u l a t i o b i l i t y of c a r r y i n g out computer experiments on the system to e l u ­ c i d a t e the e f f e c t s of v a r i o u s d e f i n a b l e c h a r a c t e r i s t i c s on the results. There remain s i g n i f i c a n t assumptions i n computer simu­ l a t i o n , mainly regarding the proper form of the conf i g u r a t i o n a l p o t e n t i a l , and i n d i v i d u a l c a l c u l a t i o n s themselves are r e l a t i v e l y lengthy undertakings i n terms of computer time. However, the i n i t i a l r e s u l t s on molecular l i q u i d s and s o l u t i o n s are encourag­ ing and are p r o v i d i n g new i n s i g h t s i n t o the s t r u c t u r a l chemistry of these systems. A l a r g e body of the computer simulation work has been r e ­ ported on model systems such as hard d i s c s , spheres or LennardJones p a r t i c l e s . Here the i n t e r p a r t i c l e p o t e n t i a l i s known and can be used t o r a p i d l y c a l c u l a t e the c o n f i g u r a t i o n a l energy of the system as required f o r Monte C a r l o s t u d i e s or the c o n f i g u r a ­ t i o n a l f o r c e on a p a r t i c l e as required f o r molecular dynamics. A great d e a l of systematic i n f o r m a t i o n has been developed from model systems which can be q u a l i t a t i v e l y a p p l i c a b l e t o r e a l sys­ tems. A s e r i e s of d e f i n i t i v e reviews of the Monte C a r l o method and r e s u l t s on model systems have been prepared by Wood-*. The dynamics approach was i n i t i a l l y c h a r a c t e r i z e d i n the s e r i e s of papers by A l d e r , Wainwright and coworkers^. A comprehensive r e ­ view of l i q u i d s t a t e theory was r e c e n t l y published by Barker and Henderson . The a p p l i c a t i o n of computer s i m u l a t i o n methods to molecular systems i s i n a r e l a t i v e l y e a r l y stage. The Monte C a r l o s t u d i e s on water by Barker and Watts (1969) and Sarkisov e t . a l . (1974) and the molecular dynamics study of water by Rahman and S t i l l i n ger (1971) ^ were the forerunners of computer s i m u l a t i o n work on chemical problems. Progress has g e n e r a l l y been slow i n t h i s area due t o the magnitude of computer f a c i l i t i e s required and the l i m ­ i t e d a v a i l a b i l i t y of p o t e n t i a l f u n c t i o n s f o r d i v e r s e chemical ap­ plications. Quite r e c e n t l y s e v e r a l computer s i m u l a t i o n s t u d i e s 7

8

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on l i q u i d water have appeared, due to L a d d , Owicki and Scheraga,12 d Swaminathan and Beveridge.^ Clementi and coworkers have reported extensive r e s u l t s on w a t e r ^ and ion-water sys­ t e m s ^ and have pioneered the use of intermolecular p o t e n t i a l functions derived from ab i n i t i o quantum mechanical c a l c u l a t i o n s . A systematic approach to the determination of a n a l y t i c a l poten­ t i a l f u n c t i o n s was contributed from t h i s Laboratory.15 The dy­ namics of ion-water i n t e r a c t i o n s f o r small c l u s t e r s has r e c e n t l y been described, and computer simulations have j u s t been reported on l i q u i d benzene, ^ l i q u i d nitrogen*? and l i q u i d ammonia. The d i l u t e aqueous s o l u t i o n of methane has been treated by Owicki and S c h e r a g a ^ and Swaminathan, Harrison and Beveridge.^ A d d i t i o n a l current a p p l i c a t i o n s to molecular l i q u i d s are i n the newly pub­ l i s h e d monograph by Watts^O d f course i n the companion papers i n t h i s volume. a

n

a

n

Q

A p p l i c a t i o n s of computer s i m u l a t i o n methods to biochemical and b i o l o g i c a l problem a l groups are i n t e r e s t e conformation based on computer s i m u l a t i o n . Clementi and co­ workers are working on water-amino a c i d p o t e n t i a l functions and 9 1 Zi

r e l a t e d problems d e a l i n g with molecular a s s e m b l i e s . Scheraea and co-workers are i n v o l v e d i n complementary studies of water*^ and methane-water s o l u t i o n , ^ and are e x p l o r i n g the use of Monte Carlo c a l c u l a t i o n s i n other problems of b i o p h y s i c a l i n t e r e s t . Karplus i s c u r r e n t l y using molecular dynamics to study p r o t e i n folding i n solution.^2 The o p p o r t u n i t i e s f o r t h e o r e t i c a l s t u d i e s of chemical and b i o l o g i c a l processes using computer s i m u l a t i o n are extremely broad and d i v e r s e , and we f u l l y expect that t h i s approach w i l l u l t i m a t e l y have a broad impact on t h e o r e t i c a l chem­ i s t r y , biochemistry and b i o l o g y . Each of the systems i n d i v i d u a l l y under c o n s i d e r a t i o n i n t h i s review has an extensive s c i e n t i f i c h i s t o r y . The background ap­ p r o p r i a t e to each system i s developed i n considerable d e t a i l i n our i n d i v i d u a l papers and, except f o r p a r t i c u l a r points and r e ­ ferences to the most recent relevant work, i s not repeated here. The focus h e r e i n i s thus on p r e s e n t a t i o n of c o l l e c t e d current Monte Carlo r e s u l t s from t h i s Laboratory obtained on a s e r i e s o f important systems, with the computational procedures and analyses c a r r i e d out on a u n i f i e d and coherent b a s i s . II.

Calculations

The methodology of Monte Carlo c a l c u l a t i o n s e s p e c i a l l y f o r p o l a r and i o n i c systems i s c u r r e n t l y an a c t i v e area of study i n computer s i m u l a t i o n theory. Aspects such as p o t e n t i a l f u n c t i o n s , boundary c o n d i t i o n s , sampling a l g o r i t h m s , convergence c r i t e r i a and t r u n c a t i o n e r r o r s are a l l r e c e i v i n g considerable research a t t e n ­ tion. The s e n s i t i v i t y of r e s u l t s to assumptions i n the c a l c u l a ­ t i o n s i s discussed p a r t i c u l a r l y i n a recent paper by Levesque, Patey and W e i s s . ^

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A l l c a l c u l a t i o n s described herein are based on s t a t i s t i c a l thermodynamic computer s i m u l a t i o n under canonical ensemble condi­ t i o n s on the system, with temperature T, number of p a r t i c l e s N, and volume V assumed and constant. Configurational integrations f o r each system are c a r r i e d out by Monte-Carlo methods i n v o l v i n g a s t o c h a s t i c walk through c o n f i g u r a t i o n space with c o n f i g u r a t i o n s s e l e c t e d on the b a s i s of t h e i r p r o b a b i l i t y i n the ensemble by the method suggested by Metropolis e t . al.^4 The N - p a r t i c l e system i s given a l i q u i d phase environment by the appropriate choice of number density and the use of image c e l l s , i . e . p e r i o d i c boundary conditions. The formalism employed i s developed f u l l y i n Refer­ ences 1 and 3. A l l c a l c u l a t i o n s unless otherwise i n d i c a t e d are based on 125 p a r t i c l e s i n a cubic c e l l at a temperature at 25°C. The volume of the c e l l i s determined by density and s p e c i f i e d i n d i v i d u a l l y f o r each system. The c a l c u l a t i o n s i n v o l v e a s p h e r i ­ c a l c u t o f f i n the p o t e n t i a l f u n c t i o n at h a l f the c e l l edge used i n conjunction with th c r i t e r i a and e r r o r bound i n the s i m u l a t i o n are developed i n terms of c o n t r o l functions as defined by Wood.5 The c o n f i g u r a t i o n a l energy of the system i n each of the Monte Carlo c a l c u l a t i o n s discussed below i s developed under the assump­ t i o n of the pairwise a d d i t i v i t y of i n t e r m o l e c u l a r i n t e r a c t i o n s by means of a n a l y t i c a l f u n c t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s of the i n t e r m o l e c u l a r i n t e r a c t i o n energy. There are of course a number of l i m i t i n g assumptions i n v o l v e d i n the c o n s t r u c t i o n of such f u n c t i o n s , such as the s i z e and s p e c i f i ­ c a t i o n of b a s i s sets i n the quantum mechanical c a l c u l a t i o n s , but they are formally w e l l defined and t h e i r c a p a b i l i t i e s and l i m i t a ­ t i o n s with respect to the pairwise i n t e r a c t i o n can be developed in d e t a i l . The f u n c t i o n s can be r e a d i l y defined on the r e q u i s i t e regions of c o n f i g u r a t i o n space, and a n i s o t r o p i c s i n the i n t e r ­ molecular i n t e r a c t i o n s are of course a u t o m a t i c a l l y i n c l u d e d . The i n d i v i d u a l terms i n the a n a l y t i c a l p o t e n t i a l f u n c t i o n should not be a s c r i b e d any p h y s i c a l s i g n i f i c a n c e ; they are simply a means f o r an i n t e r p o l a t i o n based on a l i m i t e d number of d i s c r e t e quantum-mechanically c a l c u l a t e d i n t e r m o l e c u l a r i n t e r a c t i o n energies. The major assumptions inherent i n the c o n f i g u r a t i o n a l energy c a l ­ c u l a t i o n s i n t h i s study are thus the neglect of three-body and higher order c o n t r i b u t i o n s , t r u n c a t i o n e r r o r s i n the quantum me­ c h a n i c a l c a l c u l a t i o n s of pairwise i n t e r a c t i o n energies, and s t a ­ t i s t i c a l e r r o r s i n the multidimensional curve f i t t i n g i n the ana­ l y t i c a l potential. A l l c a l c u l a t e d q u a n t i t i e s reported, with the exception of the f r e e energy f o r l i q u i d water, are based on simple ensemble averages and are produced i n a s t r a i g h t f o r w a r d manner i n the M e t r o p o l i s procedure. The a n a l y s i s of r e s u l t s i s based p a r t i c u ­ l a r l y on guasicomponent d i s t r i b u t i o n f u n c t i o n s as introduced by Ben-Nairn.25 Quasicomponent d i s t r i b u t i o n functions are defined on the s t a t i s t i c a l s t a t e of the system and give the d i s t r i b u t i o n of

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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p a r t i c l e s ( i n terms o f mole f r a c t i o n ) with any p a r t i c u l a r w e l l defined c h a r a c t e r i s t i c . The c h a r a c t e r i s t i c s r e l e v a n t to t h i s work are a) c o o r d i n a t i o n number, the number o f p a r t i c l e s w i t h i n a given radius of the center of mass of a given p a r t i c l e , and b) b i n d i n g energy, the i n t e r a c t i o n energy of a given p a r t i c l e with a l l other p a r t i c l e s i n the system. The quasicomponent d i s t r i b u ­ t i o n s are of course obtained as averages over a l l i d e n t i c a l p a r t i ­ c l e s and a l l c o n f i g u r a t i o n s o f the system. III.

L i q u i d Water

The extensive use of water as a solvent i n chemical systems, the d i r e c t p a r t i c i p a t i o n o f water i n many chemical and biochemi­ c a l processes and the unique f u n c t i o n o f water as a b i o l o g i c a l l i f e support system combine to make the s t r u c t u r e of l i q u i d water a matter of c e n t r a l importance i n understanding many chemical, biochemical and b i o l o g i c a uid water has been studie dynamics based on ad hoc models f o r the system, and there has been a long-standing controversy i n the s c i e n t i f i c l i t e r a t u r e as to whether the s t r u c t u r e of water i s best represented by a mix­ ture model, whereby the system i s viewed as a composite o f ener­ g e t i c a l l y d i s t i n c t c l u s t e r s of p o s s i b l y d i v e r s e s i z e and s t r u c ­ ture, or by an e n e r g e t i c continuum model, wherein l o c a l molecular environments with a continuous d i s t r i b u t i o n of p r o g r e s s i v e l y bent hydrogen bonds are featured. Arguments pro and con f o r each mod­ e l based on both t h e o r e t i c a l a n a l y s i s and experimental evidence are summarized i n the recent reviews by Davis and Jarzynski26 and K e l l ^ ? and general p e r s p e c t i v e on the problem i s developed i n the s e r i e s e d i t e d by F r a n k s ^ and a recent review by Gorbunov and Naberukhin.29 L i q u i d water was the f i r s t molecular l i q u i d to be e x t e n s i v e ­ l y s t u d i e d using computer s i m u l a t i o n methods. Experimental data on d i v e r s e thermodynamic p r o p e r t i e s are a v a i l a b l e and Narten, Levy and co-workers have obtained r a d i a l d i s t r i b u t i o n functions for the system by d i f f r a c t i o n methods.30 This system thus serves as a s e n s i t i v e t e s t on the q u a l i t y of the c o n f i g u r a t i o n a l poten­ t i a l used i n a computer s i m u l a t i o n . The c a l c u l a t e d oxygen-oxygen r a d i a l d i s t r i b u t i o n f u n c t i o n of l i q u i d water has been reported for q u i t e a v a r i e t y of pairwise a d d i t i v e e m p i r i c a l and a n a l y t i c a l p o t e n t i a l f u n c t i o n s . Among the e m p i r i c a l f u n c t i o n s , the ST2 pot e n t i a l ^ l i s c u r r e n t l y widely adopted. Clementi and c o - w o r k e r s ^ c a r r i e d out extensive s t u d i e s o f p a i r w i s e p o t e n t i a l f u n c t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s on the water dimer, and best agreement with the observed r a d i a l d i s ­ t r i b u t i o n f u n c t i o n was obtained using a f u n c t i o n r e p r e s e n t a t i v e of a s e l f c o n s i s t e n t f i e l d (SCF) c a l c u l a t i o n plus moderately l a r g e i n t e r m o l e c u l a r c o n f i g u r a t i o n i n t e r a c t i o n (CI) on the sys­ tem. The f u n c t i o n was reported by Matsuoka, Clementi and Y o s h i mine (MCY)32 and i s henceforth r e f e r r e d to h e r e i n as the MCY-CI

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potential. Recent s t u d i e s on l i q u i d water i n t h i s Laboratory * have been d i r e c t e d toward an a n a l y s i s of the s t r u c t u r e of the system from Monte Carlo computer based on the MCY-CI p o t e n t i a l f u n c t i o n . The c a l c u l a t i o n s were c a r r i e d out f o r 125 p a r t i c l e s at a temperature of 25°C and a volume commensurate with a density of 1 gm/cm . The ensemble averages f o r the system were based on 500K c o n f i g u r a ­ t i o n s s e l e c t e d on the b a s i s of the M e t r o p o l i s method. Conver­ gence was checked by comparing the r e s u l t s of two independent runs, one beginning at a c o n f i g u r a t i o n with r e l a t i v e l y high ener­ gy and another beginning at a c o n f i g u r a t i o n very near the minimum energy f o r the system. The c a l c u l a t i o n s converged to the same i n t e r n a l energy w i t h i n 1% and apparently no p r a c t i c a l nonergodici t i e s were encountered.33 3

The c a l c u l a t e d thermodynamic i n t e r n a l energy f o r the system obtained from Monte Carlo computer s i m u l a t i o n based on the MCYCI p o t e n t i a l was -8.58 value of -9.9 kcal/mol mainly to the assumption of pairwise a d d i t i v i t y i n the c o n f i g u r a tional potential. The c a l c u l a t e d heat c a p a c i t y , corrected f o r i n t e r n a l modes, i s 17.9 cal/mol • deg as compared with the e x p e r i ­ mental value at 25° of 18 cal/mol • deg. The c a l c u l a t e d oxygen-oxygen r a d i a l d i s t r i b u t i o n f u n c t i o n for l i q u i d water obtained from Monte Carlo computer s i m u l a t i o n based on the MCY-CI p o t e n t i a l i s shown i n F i g . 1. There i s q u i t e good accord between the c a l c u l a t e d and observed values f o r the p o s i t i o n of a l l three main peaks. In the region of the f i r s t hy­ d r a t i o n s h e l l , the shape of the c a l c u l a t e d peak agrees w e l l with experiment, although the c a l c u l a t e d maximum i s s l i g h t l y too high. For the second s h e l l , the p o s i t i o n of the maximum i s c o r r e c t but the shape i s biased towards short distance s i d e . A shoulder at ca. 3.5 8 i s c l e a r l y i n d i c a t e d . The t h i r d s h e l l appears g e n e r a l ­ ly well described. An a n a l y s i s of the s t r u c t u r e of l i q u i d water was c a r r i e d out i n terms of quasicomponent d i s t r i b u t i o n f u n c t i o n s f o r coordina­ t i o n number and b i n d i n g energy and a l s o by examining stereographi c views of s i g n i f i c a n t molecular s t r u c t u r e s . Coordination num­ ber i s c a l c u l a t e d based on R^ = 3.3, the f i r s t minimum i n the r a d i a l d i s t r i b u t i o n f u n c t i o n , and t h i s d i s t r i b u t i o n f u n c t i o n i s e s s e n t i a l l y an a n a l y s i s of the f i r s t hydration s h e l l . The c a l ­ c u l a t e d r e s u l t , d i s p l a y e d as mole f r a c t i o n of p a r t i c l e s x ( K ) vs. c o o r d i n a t i o n number K i s shown i n Figure 2. The d i s t r i b u t i o n ranges from K=2 to K=6, biased towards higher c o o r d i n a t i o n num­ bers, with K=4 predominant at 47%. The average c o o r d i n a t i o n num­ ber was found to be 4.1. The quasicomponent d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy, d i s p l a y e d as mole f r a c t i o n of p a r t i c l e s x^(is) vs. b i n d i n g energy V , i s given i n Figure 3. The d i s t r i b u t i o n i s unimodal with a maximum at -17.7 kcal/mole. As discussed by Ben-Nairn,^5 mix­ ture model f o r the system should give r i s e to a bimodal or p o l y c

a

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2.8 r

I

= ft 08 04 0 I—

m

16 2.4 3.2 4 0 4 8 5 6 6 4 7 2 8 0

R (A) —

Figure 1. Calculated points on the oxygen-oxygen radial distribution function g(R) vs. center of mass separation R for liquid water at 25 °C. Experimental data (solid line) from Narten, Danford, and Levy (SO).

0.8r

|

0.6 -

2

0.4 -

o * 0.2 0

Figure 2. Calculated quasicomponent distribution function x (K) vs. coordination number K for liquid water c

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modal d i s t r i b u t i o n of binding energies while a unimodal d i s t r i b u ­ t i o n i s c o n s i s t e n t with the idea of an e n e r g e t i c continuum model. Thus the modality of the c a l c u l a t e d d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy supports the continuum model f o r l i q u i d water. Examining these r e s u l t s from the point of view of hydrogen bond energy, an i c e - l i k e environment with four l i n e a r hydrogen bonds would be expected to have a binding energy o f ^ 2 2 kcal/mol. This energy i s on the lower edge of the c a l c u l a t e d d i s t r i b u t i o n and not h e a v i l y favored. The region of higher p r o b a b i l i t y must therefore correspond to bent hydrogen bonds. To examine t h i s point f u r t h e r , we extracted from the c a l c u l a t i o n s a number of low energy, high frequency s t r u c t u r e s . Stereographic views of two of them, chosen such that the c e n t r a l molecule has K=4, are shown i n Figures 4 and 5. There i s a notable prevalence of bent hydro­ gen bonds throughout the s t r u c t u r e . C o l l e c t i v e l y , these r e s u l t s support the e n e r g e t i c continuum model f o r l i q u i d water p r e v i o u s l y mentioned molecula other t h e o r e t i c a l work on t h i s problem, Kauzmann ^ has demon­ s t r a t e d that a 2-state mixture model f o r the system cannot formal­ l y account f o r a l l the observed data on the system. Vibrational s p e c t r a l data supporting a mixture model based on an i s o s b e s t i c point i n the Raman spectra as a f u n c t i o n of temperature have been reported by W a l r a f e n . ^ This problem has been reexamined recent­ l y by Scherer, Go and K i n t ^ who showed that the apparent isosbes­ t i c point i n Raman data that has not been decomposed i n t o iso­ t r o p i c and a n i s o t r o p i c parts i s f o r t u i t o u s . Rice and coworkers f i n d t h e i r d e t a i l e d a n a l y s i s of the 0-H s t r e t c h i n g region of the v i b r a t i o n a l s p e c t r a of amorphous s o l i d water c o n s i s t e n t with a s l i g h t l y bent hydrogen bond model. The r e l a t i o n s h i p between the observed f a r i n f a r e d spectrum of water and the molecular dynamics computer simulation r e s u l t s has been developed by Curnette and Williams. The recent r e c o n s i d e r a t i o n of s p e c t r o s c o p i c aspects of the water s t r u c t u r e problem due to Gorbunov and N a b e r u k h i n ^ f i r m l y supports the continuum model. The Monte Carlo-Metropolis method f o r computer simulation i s i d e a l l y s u i t e d to produce ensemble averages of the p r o p e r t i e s of the system. However i n the case of f r e e energy, the ensemble av­ erage expression i s not convenient f o r computational purposes due to the i l l - c o n d i t i o n e d nature of the integrand and the concomitant convergence p r o b l e m s . ^ We have r e c e n t l y c a r r i e d out a Monte Carlo c a l c u l a t i o n of the f r e e energy of l i q u i d water based on a procedure which follows from e a r l y work by Kirkwood ^ i n v o l v i n g a numerical quadrature over the i n t e r n a l energy o f the system de\eloped i n terms of an a u x i l i a r y parameter.^ The c a l c u l a t i o n was found to be computationally t r a c t a b l e and r e s u l t e d i n a Helmholz c o n f i g u r a t i o n a l free energy of -4.31- 0.07 kcal/mol compared with an observed value of -5.74 kcal/mol;^ the corresponding entropy was found to be -14.44 - 0.09 cal/deg mol against an observed val­ ue of -13.96 cal/deg mol at 25°C. The e r r o r i n free energy being 3

3

3

3 7

3 8

3

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0.08

199

r

Figure 3. Calculated quasicomponent distribution function x (v) vs. binding energy for liquid water It

v

Figure 4. Stereographic view of a fragment of a significant molecular structure contributing to the statistical state of liquid water

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of the same order of magnitude as the e r r o r i n the i n t e r n a l ener­ gy, we a s c r i b e t h i s discrepancy a l s o to the neglect higher order c o n t r i b u t i o n s to the c o n f i g u r a t i o n a l p o t e n t i a l . IV.

The D i l u t e Aqueous S o l u t i o n of Methane

The d i l u t e aqueous s o l u t i o n of methane i s a system of promi­ nent i n t e r e s t i n molecular l i q u i d s as the prototype of a non-po­ l a r molecular solute d i s s o l v e d i n l i q u i d water, and i s one of the simplest molecular systems where the hydrophobic e f f e c t ^ i s manifest. Moreover, a d e t a i l e d knowledge of the s t r u c t u r e of the methane-water s o l u t i o n at the molecular l e v e l can provide leading information on the i n t e r a c t i o n of water with d i s s o l v e d hydrocar­ bon chains i n general, and thereby c o n t r i b u t e to the t h e o r e t i c a l b a s i s f o r understanding the r o l e of water i n maintaining the s t r u c t u r a l i n t e g r i t y of b i o l o g i c a l macromolecules i n s o l u t i o n General backgroun has been r e c e n t l y reviewe important work on these systems i s due to E l e y ^ ^ d Frank and E v a n s . ^ Methane has been i d e n t i f i e d as a "structure-maker" i n aqueous s o l u t i o n i n the language of Frank and Wen.^5 The nature of s t r u c t u r a l changes i n solvent water by d i s s o l v e d hydrocarbons has f o r sometime been discussed as by T a n f o r d ^ i terms of water c l a t h r a t e formation, based on work p a r t i c u l a r l y by G l e w ^ and analogies drawn from a number of hydrate c r y s t a l s t r u c t u r e s of non-polar species, known to i n v o l v e water c l a t h r a t e s t r u c t u r e s of order 20 and 24. Key papers on t h i s t o p i c include the review by Kauzmann^? and work by Scheraga and coworkers^**. E a r l y computer simulation work on the methane-water system was reported by Dashevsky and Sarkisov.^^ Recent t h e o r e t i c a l s t u d i e s of the methane-water system are the ab i n i t i o molecular o r b i t a l c a l c u l a t i o n s of the methane-water pairwise i n t e r a c t i o n energy by Ungemach and Schaefer^O and the Monte Carlo computer s i m u l a t i o n on the d i l u t e aqueous s o l u t i o n i n the isothermali s o b a r i c ensemble by Owicki and S c h e r a g a . ^ A recent Monte Carlo study of s t r u c t u r e of the d i l u t e aque­ ous s o l u t i o n of methane from t h i s Laboratory^ i n v o l v e s one meth­ ane molecule and 124 water molecules at 25°C at l i q u i d water d e n s i t y . The c o n f i g u r a t i o n a l energy of the system i s developed under the assumption of pairwise a d d i t i v i t y using p o t e n t i a l func­ t i o n s r e p r e s e n t a t i v e of ab i n i t i o quantum mechanical c a l c u l a t i o n s f o r both the water-water and methane-water i n t e r a c t i o n s . For the water-water i n t e r a c t i o n we have c a r r i e d over the MCY-CI p o t e n t i a l f u n c t i o n used i n our previous study of the s t r u c t u r e of l i q u i d water reviewed i n the preceeding s e c t i o n . For the methane water i n t e r a c t i o n energy, we have r e c e n t l y reported-** an a n a l y t i c a l p o t e n t i a l f u n c t i o n r e p r e s e n t a t i v e of quantum mechanical c a l c u l a ­ t i o n s based on SCF c a l c u l a t i o n s and a 6-31G b a s i s s e t , with cor­ r e l a t i o n e f f e c t s included v i a second order M o l l e r - P l e s s e t (MP) corrections,52 T h i s f u n c t i o n was used f o r the methane-water cona n

n

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201

t r l b u t i o n s to the c o n f i g u r a t i o n a l p o t e n t i a l presented h e r e i n . Ensemble averages were c a l c u l a t e d on the b a s i s of a 650K stochas­ t i c walk. The c a l c u l a t e d p a r t i a l molar i n t e r n a l energy f o r t r a n s f e r of methane from gas phase to water was -23.3- 6.6 kcal/mol, compared with an experimental value of -2.6 k c a l / m o l ^ The c a l c u l a t e d r e s u l t i s seen to be negative as expected f o r the hydrophobic e f f e c t , but an order of magnitude too low. This discrepancy i s discussed f u r t h e r below. The c a l c u l a t e d r a d i a l d i s t r i b u t i o n f u n c t i o n f o r the center of mass of water molecules with respect to the methane carbon atom i s shown i n Figure 6. We f i n d a broad, unstructured f i r s t peak with a minimum i n the region o f 5.3&. I n t e g r a t i n g g(R) up to t h i s point gives an average coor­ d i n a t i o n number of 19.35. An a n a l y s i s of the s t r u c t u r e o f the d i l u t e aqueous s o l u t i o n of methane was a l s o developed i n terms of quasicomponent d i s t r i ­ bution functions and stereographi structures. The c o o r d i n a t i o was c a l c u l a t e d on the b a s i s of R = 5.38, f i x e d at the f i r s t minimum i n the methane-water r a d i a l d i s t r i b u t i o n f u n c t i o n . A p l o t of the mole f r a c t i o n of methane molecules x ( K ) vs. t h e i r corresponding water c o o r d i n a t i o n number i s given i n Figure 7. The x^(K) obtained i s a broad unimodal d i s t r i b u t i o n ranging from K=16 to K=22 with a maximum i n the region K=19 and 20, biased s l i g h t l y i n shape towards higher c o o r d i n a t i o n numbers. The c a l c u l a t e d quasicomponent d i s t r i b u t i o n f u n c t i o n f o r b i n d i n g energy, the mole f r a c t i o n of methane molecules x ( v ) as a f u n c t i o n of methane b i n d i n g e n e r g y v i s shown i n Figure 8. There i s some i n c i p i e n t s t r u c t u r e i n the curve but the e r r o r bounds on the f u n c t i o n are too l a r g e to a s c r i b e t h i s any p h y s i c a l significance. Comparing the average value of t h i s q u a n t i t y , -23.31" 6.6, with the c a l c u l a t e d p a r t i a l molar i n t e r n a l energy of t r a n s f e r f o r methane confirms that the s i g n and magnitude o f t h i s l a t t e r term are due to water s t a b i l i z a t i o n e f f e c t s . I f we now assume the ~20 kcal/mole discrepancy between the c a l c u l a t e d and observed values of the i n t e r n a l energy comes from the ~20 water molecules found i n the f i r s t h y d r a t i o n s h e l l of methane, t h i s i s an e r r o r of ~1 kcal/mole per water molecule or a f r a c t i o n of a kcal/mol per pairwise i n t e r a c t i o n or hydrogen bond. Con­ s i d e r i n g the approximations inherent i n the c a l c u l a t i o n of p a i r wise i n t e r a c t i o n energies as enumerated i n S e c t i o n I I , t h i s d i s ­ crepancy appears to be commensurate with the c a p a b i l i t i e s and l i m i t a t i o n s of the c o n f i g u r a t i o n a l energy c a l c u l a t i o n . The magnitudes and d i s t r i b u t i o n of c o o r d i n a t i o n numbers found f o r methane i n the s t a t i s t i c a l s t a t e of the d i l u t e aqueous s o l u t i o n are g e n e r a l l y c o n s i s t e n t with the presence of water c l a t h r a t e cages as shown i n Figure 9. In a number of other s t r u c ­ tures q u a s i c l a t h r a t e regions could be i d e n t i f i e d , but d e f e c t s and d i s t o r t i o n s were more prevalent than i n Figure 9. One such s t r u c t u r e , r e p r e s e n t a t i v e of other, i s given i n Figure 10. The 3

M

c

f i

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

202

COMPUTER

MODELING O F M A T T E R

Figure 5. Stereographic view of a fragment of another significant molecular structure contributing to the statistical state of liquid water

2.8 2.4

20 |

I .6

CP

- MP 0.8 0.4 0.0

—

I .8

t

24

—

32

i

—

40

i

—

i

48 R (&)

—

i

—

i

—

i

56 6.4 7 2

—

i

—

8.0

i

8.8

—

Figure 6. Calculated methane-water radial distribution gfRj vs. center of mass separation R from Monte Carlo computer simulation for the dilute aqueous solution of methane atT = 25°C

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

BEVERIDGE E T A L .

1

1

14

Structure of Liquid Water and Dilute Solutions

1 15

1 16

T 1 17

1 1 18

i1 19

i1 20

i1 21

1r — '—nI

22

23

Figure 7. Calculated quasicomponent distribution function XcfK) vs. methane coordination number K for the dilute aqueous solution of methane

v ( kcal/mol)

—

Figure 8. Calculated quasicomponent distribution function x (v) vs. methane binding energy v for the dilute aqueous solution of methane n

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER

MODELING O F M A T T E R

b Figure 9. Stereographic view of methane and its first hydration shell taken from a significant molecular structure contributing to the statistical state of the system. (Top) disposition of centers of mass of water molecules about methane (shaded) with the quasiclathrate cage delineated; (bottom) disposition of water molecules about methane in the same structure.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16. BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions

b Figure 10. Stereographic view of methane and its first hydration shell taken from another significant molecular structure contributing to the statistical state of the system. (Top) disposition of centers of mass of water molecules about methane (shaded) with the quasiclathrate cage delineated; (bottom) disposition of water molecules about methane in the same structure.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

205

206

COMPUTER MODELING OF

MATTER

emergent d e s c r i p t i o n of the aqueous s o l u t i o n environment of meth­ ane i s that of a d i s t o r t e d and d e f e c t i v e continuum c l a t h r a t e structure. E f f e c t s of s o l u t e methane on solvent water s t r u c t u r e were developed i n terms of the d i f f e r e n c e i n Xg(K) and x^(i/) c a l c u ­ l a t e d f o r the s o l u t i o n and f o r the pure l i q u i d . The e f f e c t s of bulk water are removed by the d i f f e r e n c i n g , allowing the d i r e c t d i s p l a y of s t r u c t u r a l changes i n solvent water. The d i f f e r e n c e p l o t s r e v e a l under high s t a t i s t i c a l e r r o r an increase i n 4-coordinate species and a s l i g h t but c l e a r l y discernable s h i f t t o ­ ward lower binding energy f o r the solvent molecules, p r o v i s i o n a l ­ l y consistent with general ideas of " s t r u c t u r e making". F u r t h e r work i s c u r r e n t l y i n progress on t h i s point. V.

D i l u t e Aqueous Solutions

of Monatomic Cations and

Anions.

The p a r t i c u l a r s i g n i f i c a n c chemistry makes the s t r u c t u r monatomic cations and anions a l s o a t o p i c of fundamental i n t e r ­ e s t . Moreover, the sodium and potassium ions i n p a r t i c u l a r f i g ­ ure prominently i n biochemical membrane p o t e n t i a l phenomena, and t h e i r hydration s t a t e i n aqueous s o l u t i o n i s an important f a c t o r i n ion s e l e c t i v i t y and membrane permeability i n b i o l o g i c a l sys­ tems. Modern t h e o r e t i c a l studies of these systems date from the c l a s s i c paper of Bernal and Fowler i n 1933. ^ The current s t a t e of both experimental and t h e o r e t i c a l research on i o n i c s o l u t i o n s i s the subject of s e v e r a l recent comprehensive reviews p a r t i c u ­ l a r l y by Friedman and c o - w o r k e r s . The prevalent d e s c r i p t i v e ideas about the l o c a l aqueous s o l u t i o n environment of ions stems from the work of Frank and Wen,^5 h o p a r t i t i o n e d the solvent i n ­ to three regions. In the immediate v i c i n i t y of the i o n , region A, water molecules are t i g h t l y bound and h i g h l y o r i e n t e d . Region C at l a r g e distance from the ion was considered as e s s e n t i a l l y bulk water, and the i n t e r v e n i n g region B i s a region of s t r u c ­ t u r a l ambiguity i n t e r f a c i n g regions A and B. Recently Kistenmacher, Popkie and dementi *' reported analy­ t i c a l intermolecular p o t e n t i a l functions f o r ion-water i n t e r a c ­ t i o n s r e p r e s e n t a t i v e of near Hartree-Fock molecular o r b i t a l c a l ­ culations. These functions along with water-water p o t e n t i a l s of commensurate q u a l i t y were ised to study the s t r u c t u r e and s o l v a ­ t i o n numbers of ion-water c l u s t e r s based on energy o p t i m i z a t i o n . Monte Carlo simulation work based on these functions have been reported by Mruzik, Abraham, Scheiber and P o u n d focussing on free energy c a l c u l a t i o n s , and by Watts^O d coworkers on ion p a i r s i n large water c l u s t e r s . McDonald and Rasaiah have c a r r i e d out very large order studies of ion p a i r s i n s o l u t i o n based on model p o t e n t i a l f u n c t i o n s . Molecular dynamics on i o n water systems have been reported by Briant and B u r t o n ^ and Heinzinger and Vogel.^0 5

55

w

5

57

a n

5

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

BEVERIDGE E T A L .

Structure of Liquid Water and Dilute Solutions

207

12 0

10.0

|

8.0

_

6.0

&

4.0

R ( £]

Figure 11. Calculated cations-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of lithium atT = 25°C

Figure 12. Calculated cation—water radial distribution function vs. center of mass separation R for the dilute aqueous solution of sodium atT = 25° C

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

208

COMPUTER MODELING O F M A T T E R

Figure 13. Calculated cation-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of potassium at T = 25°C

4.8

r

4.0

!" _ 2.4 cr ^

I .6

— 0.8 0.0 08

. v I

I

1.6 2 4

^ ^ ^ / N ^ -

1

F

' 1

1

3.2 4 0

48

1

1

1

5 6 6.4 7.2

1

8.0

R (&)

Figure 14. Calculated anion-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of fluoride at T = 25°C

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions 209

2.8 2.4 2.0

I

1 6

5

I 2

cn

* 0.8

•

Xs/KT

V

0.4

o.o | 0.8

I | i 1 1 1 1 1 1 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

R [&)—+>

Figure 15. Calculated anion-water radial distribution function vs. center of mass separation R for the dilute aqueous solution of chloride at T = 25°C

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

210

COMPUTER

08 0.6

R

MODELING OF

MATTER

= 3 30 &

M

(X

(K))

c

=6 77 ± 0 . 1 8

0 4 0.2 0.0 0.8

t

1

Xo

0.6

R =3 M

00&

(X (K))

-- 5 9 8 ± 0

C

22

0.4 0.2 "^"]

00

1 0 0.8

-i

R = M

2.65

1

—

No —

4

r

I

( X ( K ) ) =5 0 5 + 0 0 7 C

0.6 0.4 0.2 Li* 0.0

1

T"

4 K

5 —

Figure 16. Calculated quasicomponent distribution functions x fK) vs. ion coordination number K for dilute aqueous solutions of alkali metal cations r

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16.

BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions

211

1.0 0.8

R

=4.10 &

M

(x

c

(K)) =7.69 ± 0 . 2 9

0.6 0.4 0.2 cr

| 0.0

1

1

1

1—

h—

R =3.30 £ M

| .0

<X (K)) = 5 . 4 6 ± 0 . 2 4 C

x 0.8 0.6 0.4 0.2 0.0

F~ —I——1~

3

4

i

i

5 K

6

1 9

r~

1

10 11

Figure 17. Calculated quasicomponent distribution functions x (K) vs. ion coordination number K for dilute aqueous solutions of halide anions c

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

212

COMPUTER

0.050

MODELING O F M A T T E R

A

0025 0.0

^ cn x

1

1

1

1

I

|

1

1

1

1

I

1

0050 0025 0.0

1

1

0.050 0 025 0 0 -300

-250

-275

-225

175

-200

-150

v ( kcal /mol)

Figure 18. Calculated quasicomponent distribution functions x (v) vs. ion binding energy for dilute aqueous solutions for alkali metal cations B

v

0050 0025 0.0

V , c r 1

1

1

-250

-225

i

i

i

-175

-150

0.050 0.025

V

0.0 -275

-200

v ( kcal /mol) —

Figure 19. Calculated quasicomponent distribution functions Xn(v) vs. ion binding energy for dilute aqueous solutions for halide anions v

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16.

BEVERIDGE E T A L .

Structure of Liquid Water and Dilute Solutions 213

Figure 20. Stereographic view of a significant molecular structure contributing to the statistical state of the dilute aqueous solution of KS

Figure 21. Stereographic view of a significant molecular structure contributing to the statistical state of the dilute aqueous solution of F~

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

214

COMPUTER MODELING O F M A T T E R

We have r e c e n t l y c a r r i e d out Monte Carlo computer simulation of d i l u t e aqueous s o l u t i o n s of the monatomic cations L i , Na and K and the monatomic anions F~" and C l ~ using the KPC-HF functions f o r the ion-water i n t e r a c t i o n and the MCY-CI p o t e n t i a l f o r the water-water i n t e r a c t i o n . The temperature of the systems was taken to be 25° and the density chosen to be commensurate with the p a r t i a l molar volumes as reported by M i l l e r o . ^ l The c a l c u l a t ­ ed average q u a n t i t i e s are based on from 600- 900K c o n f i g u r a t i o n s a f t e r e q u i l i b r a t i o n of the systems. The c a l c u l a t e d ion-water r a d i a l d i s t r i b u t i o n functions are given f o r the d i l u t e aqueous s o l u t i o n s o f L i , K , Na+, F~ and C I " i n Figures 11-15, respec­ tively. An a n a l y s i s of the s t r u c t u r e of the d i l u t e aqueous s o l u t i o n s of monatomic cations and anions was developed i n a manner analo­ gous to that presented above f o r pure water and the d i l u t e aque­ ous s o l u t i o n of methane. The d i s t r i b u t i o n of coordination num­ bers x^,(K) vs. K i s presente f o r the anions i n Figur and centered i n the region K=5, 6 and 7. The average water co­ o r d i n a t i o n number of each ion i s found by determining the i n t e ­ g r a l of the f i r s t peak i n the ion-water r a d i a l d i s t r i b u t i o n func­ t i o n , and these values are also recorded on Figures 16 and 17. The binding energy analyses are given i n Figures 18 and 19 f o r cations and anions r e s p e c t i v e l y ; each one i s continuous and u n i modal. Representative stereographic s t r u c t u r e s are given f o r the c a t i o n K i n Figure 20 and f o r the anion F~ i n Figure 21; the other cations and anions introduce no e s s e n t i a l l y new q u a l i t a t i v e features of the s t r u c t u r e . These r e s u l t s should be considered i n the context of observ­ ed discrepancy of 10-40% between the c a l c u l a t e d and observed i n t e r n a l energy o f t r a n s f e r due to the assumption o f pairwise a d d i t i v i t y i n the c o n f i g u r a t i o n a l p o t e n t i a l and the t r u n c a t i o n o f the p o t e n t i a l i n c o n f i g u r a t i o n a l energy c a l c u l a t i o n s . The e f f e c t of higher order terms i n the c o n f i g u r a t i o n a l p o t e n t i a l on the a n a l y s i s o f s t r u c t u r e as presented h e r e i n has yet to be determin­ ed. +

+

+

+

VI.

Summary and Conclusions

The s e r i e s of studies of molecular l i q u i d s presented h e r e i n c o l l e c t r e s u l t s on a diverse s e t of chemically relevant systems from a uniform t h e o r e t i c a l point of view: ab i n i t i o c l a s s i c a l s t a t i s t i c a l mechanics on the (T,V,N) ensemble with p o t e n t i a l functions representative of ab i n i t i o quantum mechanical c a l c u ­ l a t i o n s of pairwise i n t e r a c t i o n s and s t r u c t u r a l a n a l y s i s c a r r i e d out i n terms of quasicomponent d i s t r i b u t i o n f u n c t i o n s . The l e v e l of agreement between c a l c u l a t e d and observed q u a n t i t i e s i s quoted to i n d i c a t e the c a p a b i l i t i e s and l i m i t a t i o n s to be expect­ ed o f these c a l c u l a t i o n s and i n that perspective we f i n d a num­ ber of s t r u c t u r a l features of the systems p r e v i o u s l y discussed on

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16.

BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions

215

an e m p i r i c a l b a s i s to be q u a n t i f i e d . The present r e s u l t s point p a r t i c u l a r l y to a) a continuum model f o r l i q u i d water, b) a d i s ­ t o r t e d , d e f e c t i v e c l a t h r a t e environment f o r methane i n aqueous s o l u t i o n , and c) d i s t i n c t l y d i f f e r e n t d i s t r i b u t i o n s o f e q u i l i b ­ rium c o o r d i n a t i o n numbers f o r N a and K i n water which may be of s i g n i f i c a n c e i n biomembrane phenomena. U l t i m a t e l y i t i s hoped that a d e s c r i p t i v e chemistry of molecular l i q u i d s u s e f u l i n d i v e r s e a p p l i c a t i o n s can be developed on a r i g o r o u s b a s i s u s i n g the r e s u l t s of computer s i m u l a t i o n s and we c o n t r i b u t e these s t u d ­ i e s as a step i n that d i r e c t i o n . +

+

VII. Acknowledgement Support f o r t h i s research comes from NIH Grant //1-R01NS12149-03 from the N a t i o n a l I n s t i t u t e s o f N e u r o l o g i c a l and Com­ municative Diseases and Stroke and a CUNY F a c u l t y Research Award. D.L.B. acknowledges U.S.P.H.S 6TK04-GM21281 from th Studies.

VIII. References 1. 2. 3. 4. 5.

6.

7. 8. 9. 10. 11. 12. 13.

S. Swaminathan, and D.L. Beveridge, J . Am. Chem. Soc., 99, 8392 (1977). M. Mezei, S. Swaminathan, and D.L. Beveridge, J . Am. Chem. Soc., 100, 3255 (1978). S. Swaminathan, S.W. H a r r i s o n , and D.L. Beveridge, J . Am. Chem. Soc., in press. M. Mezei and D.L. Beveridge, MS in preparation. W.W. Wood and J . J . Erpenbeck, Am. Rev. Phys. Chem., 27, 319 (1976). J . J . Erpenbeck and W.W. Wood, in "Statistical Mechanics, Pt B", B.J. Berne, ed., Plenum Press, New York, N.Y. 1977 p 1. B.J. A l d e r and T.E. Wainwright, J . Chem. Phys., 31, 459 (1959); 33, 1439 (1960). B.J. A l d e r , D.M. Gass, and T.E. Wainwright, J . Chem. Phys., 53, 3013 (1970). J.A. Barker and D. Henderson, Rev. Mod. Phys., 48, 587 (1976). J.A. Barker and R.O. Watts, Chem. Phys. L e t t . , 3, 144 (1969). G.N. Sarkisov, V.G. Dashevsky, and G.G. Malenkov, Mol. Phys. 27, 1249 (1974). A. Rahman and F.H. Stillinger, J. Chem. Phys., 55, 336 (1971). A.J.C. Ladd, Mol. Phys., 33, 1039 (1977). J.C. Owicki and H.A. Scheraga, J . Am. Chem. S o c . , 99, 7403 (1977). E. Clementi, " L i q u i d Water S t r u c t u r e " , Springer V e r l a g , New York (1976).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

216

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MODELING

OF MATTER

14. J . Fromm, E. Clementi, anf R.O. Watts, J . Chem. Phys., 62, 1388 (1975). E. Clementi, R. B a r s o t t i , J . Fromm and R.O. Watts, Theoret. Chem. Acta (Berl.) 43, 101 (1976). 15. S. Swaminathan, R.J. Whitehead, E. Guth and D.L. Beveridge, J . Am. Chem. Soc., 99, 7817 (1977). 16. D.J. Evans and R.O. Watts, Mol. Phys., 32, 93 (1976). 17. P.S.Y. Cheung and J.G. Powles, Mol. Phys., 32, 1383 (1976). 18. I.R. McDonald and M.L. K l e i n , J . Chem. Phys., 64, 1790 (1976); P.K. Mehrotra and D.L. Beveridge, MS. i n p r e p a r a t i o n . 19. J.C. Owicki and H.A Scheraga, J . Am. Chem. Soc., 99, 7413 (1977). 20. R.O. Watts, " L i q u i d State Chemical Physics", John Wiley & Sons, New York (1976). 21. E. Clementi, F. L a v a l l o n e , and R. Scordamoglia J . Am. Chem. Soc., 99, 5531 (1977); R. Scordamoglia, F. Cavallone, and E. Clementi, J . Am Chem Soc. 99 5545 (1977); G B o l i s and E. Clementi, J . Am 22. P. Rossky, M. Karplus tion. 23. D. Levesque, G.N. Patey and J . J . Weis, Mol. Phys. 34, 1077 1091 (1977). 24. N.A. M e t r o p o l i s , A.W. Rosenbluth, M.N. Rosenbluth, A.H. Tell­ er and E. Teller, J. Chem. Phys., 21, 1087 (1953). 25. A. Ben-Nairn, "Water and Aqueous S o l u t i o n s " , Plenum Press, New York, N.Y. 1974. 26. C.M. Davis and J . J a r z y n s k i i n "Water and Aqueous S o l u t i o n s " R.A. Home, ed., John Wiley and Sons, New York, N.Y. 1972. The mixture model emerged from the earliest thoughts about the s t r u c t u r e of liquid water as a composite of ice-like (bulky) c o n t r i b u t i o n s i n e q u i l i b r i u m with a dense component. Highly developed forms of t h i s theory are the interstitial model, (O.Ya. Samoilov, Zh. Fiz. Khim. 20, 1411 (1946). V.A. Mikhailov, Zh. S t r u c t . Khim. 8, 189 (1967), 9, 397 (1968); A.H. Narten and H.A. Levy, Science 165, 3892 (1969). and the f l i c k e r i n g c l u s t e r model, (A.T. Hagler, H.A. Scherga and G. Nemethy, Ann. N.Y. Acad. S c i . 204, 51 (1973). 27. G.S. K e l l i n "Water and Aqueous S o l u t i o n s " R.A. Home, ed., John Wiley and Sons, New York, N.Y. 1972. The e n e r g e t i c continuum model f o r liquid water was developed during d i s ­ s e r t a t i o n research by J.A. Pople, Proc. Roy. Soc. (London), Ser A, 205, 163 (1951). 28. F. Franks, ed., "Water-A Comprehensive T r e a t i s e " V o l s . I - I I I , Plenum Press, New York, N.Y. 1972-73. 29. B.Z. Gorbunov and Yu.I. Naberukhin, J . S t r u c t . Chem. 16, 703 (1975). 30. A.H. Narten and H.A. Levy, J . Chem. Phys., 55, 2263 (1971). A.H. Narten, J . Chem. Phys., 55, 5681 (1972). 31. F.H. Stillinger and A. Rahman, J . Chem. Phys., 60, 1545 (1974).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

16.

BEVERIDGE ET AL.

Structure of Liquid Water and Dilute Solutions

217

32. O. Matsuoka, E. Clementi, and M. Yoshimine, J . Chem. Phys., 64, 1351 (1976). 33. B. Larsen and S.A. Rodge, J . Chem. Phys., 68, 1309 (1978). C. P a n g a l i , M. Rao, and B.J. Berne, Chem. Phys. L e t t . , in press. 34. W. Kauzmann, Colloqes I n t e r n a t i o n a l du C.N.R.S. #2461, 63 (1976). 35. G. Walrafen i n "Water - A Comprehensive T r e a t i s e " ,Vol.I, F. Franks, ed., Plenum Press, New York. 1972, p.151. 36. J.R. Scherer, M.K. Go and S. K i n t , J . Phys. Chem. 78, 1304 (1974). 37. S.A. Rice, W.G. Madden, R. McGraw, M.G. Sceats, and M.S. Berger, J . Chem. Phys. i n p r e s s . 38. B. Curnutte and D. W i l l i a m s , i n S t r u c t u r e of Water and Aqueous Soln, W.A.B. Luck, ed. ISBN 3-527-25588-5 (1974). 39. J.G. Kirkwood, i n "Theory of L i q u i d s " B.J A l d e r Ed. Gor­ don and Breach, Ne 40. C. Tanford, "The Hydrophobi New York, N.Y. 1973. 41. F. Franks i n "Water-A Comprehensive T r e a t i s e " , I I , F. Franks ed. Plenum Press, New York,N.Y.1972,p.l. 42. A Ben-Naim i n "Water and Aqueous S o l u t i o n , " R.A. Horne, ed., W i l e y - I n t e r s c i e n c e , New York, N.Y. 1972, p. 425. 43. D.D. Eley, Trans. Faraday Soc., 35., 1281 (1939). 44. H.S. Frank and M.W. Evans, J . Chem. Phys., 13, 507 (1945). 45. H.S. Frank and W.Y. Wen, Disc. Faraday Soc. 24, 133 (1957). 46. D.N. Glew, J . Am. Chem. Soc., 66, 605 (1962). 47. W. Kauzmann, Adv. P r o t e i n Chem., 14, 1 (1959). 48. G. Nemethy and H.A. Scheraga, J . Chem. Phys. 36, 3382, 3401 (1962). 49. V.G. Dashevsky and G.N. S a r k i s o v . , Mol. Phys., 27, 1271 (1974). 50. S.R. Ungemach and H.F. Schaefer I I I , J . Am. Chem. Soc., 96, 7898 (1974). 51. S.W. H a r r i s o n , S. Swaminathan, D.L. Beveridge and R. D i t c h f i e l d Int. J . Quantum Chem., i n press. 52. J.A. Pople, J.S. B i n k l e y , and R. Seeger, I n t . J . Quantum Chem., Symp. No. 10, 1 (1976). 53. M. Yaacobi and A. Ben-Nairn, J . Phys. Chem., 78, 175 (1924); H. Edelnoch and J.C. Osborne, J r . , Advan. P r o t e i n Chem., 30, 188 (1976). 54. J.D. Bernal and J . Fowler, J . Chem. Phys., 1, 515 (1933). 55. H.L. Friedman and W.D.T. Dale i n "Modern T h e o r e t i c a l Chemis­ try", Vol IV, Statistical Mechanics, pt A, B.J. Berne, ed., Plenum Press, New York, N.Y. 1977. 56. H. Kistenmacher, H. Popkie and E. Clementi, J . Chem. Phys. 61, 799 (1974). 57. M.R. Mruzik, F.F. Abraham, D.E. S c h r e i b e r , and G.M. Pound, J . Chem. Phys., 64, 481 (1976).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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58. 59.

I.R. McDonald, J.C. Rasaiah Chem. Phys. L e t t . 34, 382. (1975). C.L. B r i a n t and J . J . Burton, J . Chem. Phys., 60, 2849 (1974). K. Heinzinger and P.C. Vogel, Z. N a t u r f o r s c h . , 29a, 1164 (1974); P.C. Vogel and K. Heinzinger, Z. N a t u r f o r s c h . , 30a, 789 (1975). F.J. M i l l e r o i n "Water and Aqueous S o l u t i o n " , R.A. Horne, ed., W i l e y - I n t e r s c i e n c e , New York, N.Y. 1972, p. 519.

60.

61.

RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

17 Computer Modeling of Quantum Liquids and Crystals M. H. KALOS, P. A. WHITLOCK, and D. M. CEPERLEY Courant Institute of Mathematical Sciences, 251 Mercer Street, New York, NY 10012

There are a number of many-body systems which exhibit quantum effects on a macroscopic scale. These include liquid and crystal states of both He-3 and He4, the electron gas, and neutron matter which probably constitutes the interior of pulsors. In addition, "nuclear matter" - a hypothetical extensive system of nucleons has been studied for the insight one may gain into the nature of finite nuclei. The theoretical studies of these systems have by now a long history, but are by no means concluded. In the last few years, significant advances have been made. This has come in part from the maturity of and gradual unification of many-body theory, in part from the development and application of powerful new expansion procedures, especially varieties of hypernetted-chain equations (1) and finally to the growing power of computer simulation methods for quantum systems. This article is intended as a review of some recent development in computational methods for extensive quantum systems, and of the relation between results so obtained to the evolution of other theoretical work. C o m p u t a t i o n a l m o d e l l i n g o f quantum many-body s y s tems i s n o t e s p e c i a l l y n o v e l . The e a r l y h i s t o r y o f M o n t e C a r l o m e t h o d s i n c l u d e d many p r o p o s a l s f o r t h e solution of Schrodinger s equation with intended a p p l i c a t i o n t o t h e many-body p r o b l e m . Unfortunately, the c o m p u t a t i o n a l p o w e r a v a i l a b l e was n o t a d e q u a t e t o d o more t h a n s i m p l e e x e r c i s e s . T h e f i r s t work i n w h i c h a s i g n i f i c a n t c o n t r i b u t i o n t o t h e o r y was made was t h a t o f W. L . M c M i l l a n (2_) who n o t e d t h a t t h e g e n e r a l s a m p l i n g a l g o r i t h m o f M e t r o p o l i s e t a l . (3)developed to t r e a t e q u i l i b r i u m c h e m i c a l systems c o u l d be used e q u a l l y w e l l t o o b t a i n v a r i a t i o n a l estimates o f the energy o f a m a n y - b o d y s y s t e m when t h e t r i a l f u n c t i o n h a s a product form. S i n c e t h e n , a l a r g e number o f s i m i l a r 1

0-8412-0463-2/78/47-086-219$05.00/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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MODELING O F M A T T E R

c a l c u l a t i o n s have been c a r r i e d o u t t r e a t i n g extensive s y s t e m s o f atoms f r o m h y d r o g e n t o n e o n . For a thorough r e v i e w o f t h e s e c a l c u l a t i o n s see r e f e r e n c e ( £ ) . I t i s l i k e l y t h a t c a l c u l a t i o n s o f t h i s k i n d w i l l be e v e n more u s e d i n t h e f u t u r e s i n c e t h e y a r e w e l l s u i t e d f o r modern m i n i c o m p u t e r s . We w o u l d l i k e t o e m p h a s i z e h e r e some a d d i t i o n a l m e t h o d o l o g i c a l developments and t h e i r r e s u l t s . The f i r s t i s the v a r i a t i o n a l treatment of f u l l y antisymmetrized t r i a l functions (5). The second i s the G r e e n ' s f u n c t i o n Monte C a r l o a l g o r i t h m ( £ , 6) w h i c h h a s , i n e f f e c t , made p o s s i b l e t h e n u m e r i c a l i n t e g r a t i o n of the S c h r o d i n g e r e q u a t i o n . F e r m i o n Monte C a r l o Consider

a

Hamiltonia

H = - ^ X Z v

i

+

1

Z I v ( r ±<J

)

.

(1)

L e t R s t a n d f o r the c o o r d i n a t e s o f a l l p a r t i c l e s and ^
T

EQ b e i n g the energy McMillan noted that

of if

2

T

T

i f a population of from a p r o b a b i l i t y P (R)

Then

the

- 1

over

> E

,

|^ (R)| dR 2

T

T

p o i n t s {R } w e r e d r a w n density function m

2

T

average

2

H^ (R)dR/

=|* (R)| /|

T

(R)| dR

(2)

the ground s t a t e o f the system. E q . (2) w e r e r e w r i t t e n as

|* (R)| * (R) and dom

|^

**(R)H* (R)dR/

T

the

at

ran-

(4)

|* (R)| dR 2

T

(3)

population of

the

take

have

quantity

m=l i s Eip. F u r t h e r m o r e duct form: * (R,A) T

if

we

="TTf(r..,A) i<j 1

J

ip (R) T

= exp{z

the

pro-

u ( r . . , A ) } (6)

\ YZ i<j

then sampling p i s exactly analagous Boltzmann d i s t r i b u t i o n s i n c e u(r^j,A) T

to

J

to is

sampling the a repulsive

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

17.

KALOS ET AL.

Quantum Liquids and Crystals

221

f u n c t i o n w h i c h s e r v e s t o keep t h e p a r t i c l e s a p a r t s i m ­ i l a r t o the r o l e o f t h e p o t e n t i a l , v ( r ^ j ) , f o r c l a s s i ­ c a l s y s t e m s . The s a m p l i n g may be a c c o m p l i s h e d by t h e Monte C a r l o method o f M e t r o p o l i s e t a l . (3^)- T h i s method i s a u s e f u l s i m u l a t i o n method f o r Bose l i q u i d s l i k e He-4. C r y s t a l s can a l s o be s t u d i e d w i t h a t r i a l f u n c t i o n o f t h e form *

(R) = exp{-

\

u ( r . .)} TT * (

r

J

(7)

where (r) i s a s i n g l e p a r t i c l e o r b i t a l w h i c h l o c a l ­ i z e s p a r t i c l e I c l o s e t o l a t t i c e s i t e s^. U s u a l l y <(> i s s e t as 2


(8)

As w i t h o t h e r v a r i a t i o n a mized w i t h r e s p e c t parameter The c a l c u l a t i o n of p r o p e r t i e s of f e r m i o n systems was a c c o m p l i s h e d by s i m u l a t i n g t h e c o r r e s p o n d i n g Bose system and e s t i m a t i n g t h e e f f e c t o f a n t i s y m m e t r y by an e x p a n s i o n p r o c e d u r e due t o Wu and Feenberg (7^) and e x t e n d e d by S c h i f f and V e r l e t (8). I n the l a t t e r , i t i s assumed t h a t a f e r m i o n wave f u n c t i o n i s *F

( R )

=

TT

e x

[

P -

I

u(r

i : J

) ] d e t { < D ( r , a ) } (9) £

k

k

where a r e p l a n e wave o r b i t a l s t i m e s s p i n e i g e n f u n c t i o n s . Then one expands t h e energy e x p e c t a t i o n i n s u c c e s s i v e p e r m u t a t i o n s d e r i v i n g from the d e t e r m i n a n t . The f i r s t o r d e r o f c o r r e c t i o n comes from p a i r permu­ tations. S c h i f f and V e r l e t m i n i m i z e d the c o r r e c t e d f e r m i o n energy o f l i q u i d s t a t e s o f He-3 w i t h r e s p e c t t o p a r a m e t e r s i n u, f i n d i n g a p p a r e n t l y good c o n v e r ­ gence o f t h e Wu-Feenberg e x p a n s i o n . These c a l c u l a ­ t i o n s used a L e n n a r d - J o n e s p o t e n t i a l : v(r)

= 4e[ ( a / r )

1 2

-(a/r) ]; 6

e = 10.22°K, a = 2.556A . (10)

C e p e r l e y , C h e s t e r , and K a l o s (5) showed t h a t w i t h a t r i a l f u n c t i o n o f t h e form g i v e n Ey Eq. ( 9 ) , t h e M e t r o p o l i s a l g o r i t h m c o u l d be c a r r i e d o u t d i r e c t l y t o sample 11|> | i n a way t h a t was c o m p u t a t i o n a l l y econom­ i c a l i n s p i t e o f the n e c e s s i t y o f e v a l u a t i n g s e v e r a l d e t e r m i n a n t s a t each s t e p of t h e random w a l k . They found t h a t the c o n v e r g e n c e o f t h e p e r m u t a t i o n expan­ s i o n appears s a t i s f a c t o r y a t o n l y one d e n s i t y - t h a t for which the pressure i s zero. At other d e n s i t i e s ,

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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MODELING O F M A T T E R

or for the k i n e t i c or p o t e n t i a l energies separately, the convergence i s poor. F o r e x a m p l e a t pcP = 0 . 4 1 4 , t h e f e r m i o n Monte C a r l o g i v e s an e n e r g y o f 2 . 8 4 ° K p e r atom w h i l e t h e f i r s t o r d e r p e r m u t a t i o n e x p a n s i o n g i v e s 1.77°K. In fact the f i r s t order gives only h a l f the f u l l antisymmetrization correction. I t s h o u l d be s t r e s s e d t h a t t h e s e c a l c u l a t i o n s s t i l l do n o t g i v e c o r r e c t l y the observed energy of l i q u i d He-3 ( - 1 . 3 ° K instead of - 2 . 5 ° K ; of course there i s considerable c a n c e l l a t i o n ; the p o t e n t i a l energy i s about - 1 1 ° K per atom). T h i s c a n be a s c r i b e d t o an i n c o r r e c t p o t e n t i a l b u t t h e r e i s e v e n more s e r i o u s d o u b t a b o u t t h e a c c u r a cy o f the p r o d u c t t r i a l f u n c t i o n . We w i l l d i s c u s s i n t h e n e x t s e c t i o n some c o n s e q u e n c e s o f t h e n e g l e c t o f t h r e e body c o r r e l a t i o n s i n the t r i a l f u n c t i o n . R e f e r e n c e (_5) a l s o t r e a t e d f e r m i o n s y s t e m s w h i c h model n e u t r o n and n u c l e a plified pair potential c o m b i n a t i o n o f Yukawa f u n c t i o n s : v(r)

(ID l

The W u - F e e n b e r g e x p a n s i o n a l w a y s u n d e r e s t i m a t e d t h e e n e r g y c a l c u l a t e d by the f e r m i o n Monte C a r l o method. In t r e a t i n g c r y s t a l phases of f e r m i o n systems i t is known t h a t t h e e f f e c t o f a n t i s y m m e t r y ( t h e e x c h a n g e energy) i s very s m a l l . E q s . (7) a n d (8) may t h e n b e used for the t r i a l f u n c t i o n s . The e q u a t i o n o f s t a t e of fermions i n t e r a c t i n g w i t h a s i n g l e r e p u l s i v e Yukawa was d e t e r m i n e d f o r b o t h l i q u i d a n d c r y s t a l p h a s e s and a c r i t i c a l s t r e n g t h d e t e r m i n e d f o r t h e existence of a phase t r a n s i t i o n . It i s also worth mentioning that r e c e n t l y d e v e l o p e d i n t e g r a l e q u a t i o n s (9_, !10) t h a t e x t e n d t h e HNC m e t h o d t o i n c l u d e a n t i s y m m e t r i z a t i o n ( t h e FHNC e q u a t i o n s ) g i v e a good a c c o u n t o f the n e u t r o n f l u i d energ i e s a t low d e n s i t i e s . At high d e n s i t i e s , different e x p r e s s i o n s f o r t h e k i n e t i c e n e r g y t h a t w o u l d be e q u a l in a correct variational calculation give rather d i f f e r e n t answers i n d i c a t i n g t h a t the expansions are not w e l l behaved. I n t e r e s t i n g l y , one o f t h e e x p r e s s i o n s f o r t h e k i n e t i c e n e r g y , t h a t due t o P a n d h a r i pande and Bethe (1), g i v e s good agreement w i t h the f e r m i o n Monte C a r l o r e s u l t s i n a l l c a s e s . This fact i s not yet e x p l a i n e d . C e p e r l e y (11) has a p p l i e d the methods d i s c u s s e d h e r e t o t h e t r e a t m e n t o f a n e l e c t r o n g a s i n b o t h two and t h r e e d i m e n s i o n s w i t h a u n i f o r m n e u t r a l i z i n g b a c k ground. He c o n s i d e r e d t h r e e p o s s i b l e s t a t e s : fluid with h a l f the spins up; f l u i d with a l l spins a l i g n e d ;

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

17.

Quantum Liquids and Crystals

KALOS E T A L .

223

and c r y s t a l p h a s e s . He found t h a t a t low d e n s i t y t h e c r y s t a l phase i s f a v o r e d . A t i n t e r m e d i a t e d e n s i t i e s the t o t a l l y p o l a r i z e d f l u i d i s most s t a b l e and a t h i g h d e n s i t i e s theequilibrium state i s that of the unpolarized f l u i d . I n t h r e e d i m e n s i o n s t h e t r a n s i t i o n den­ s i t i e s o c c u r a t (5.4±0.4) x 1 0 / c m and a t (9.2±1.8) x lol^/cm . fi d good agreement f o r t h e u n p o l a r i z e d f l u i d w i t h o t h e r t h e o r e t i c a l work (!L2, 1J3) a t low d e n s i t i e s b u t n o t a t h i g h d e n s i t y where t h e Monte C a r ­ l o i s a t i t s most a c c u r a t e . The e q u a t i o n o f s t a t e f o r the c r y s t a l i s i n agreement w i t h an anharmonic expan­ s i o n method (.14) . We b e l i e v e t h a t f e r m i o n Monte C a r l o w i l l f i n d s i g ­ n i f i c a n t f u t u r e a p p l i c a t i o n s , p o s s i b l y i n Quantum Chemistry. 1 3

3

3

H

e

n

s

Green's F u n c t i o n Mont T h i s i s a c l a s s o f a l g o r i t h m s w h i c h makes f e a s i b l e on c o n t e m p o r a r y computers an e x a c t Monte C a r l o s o l u ­ t i o n o f the Schrodinger equation. I t i s exact i n the sense t h a t a s t h e number o f s t e p s o f t h e random w a l k becomes l a r g e t h e computed energy t e n d s toward t h e ground s t a t e energy o f a f i n i t e system o f bosons. I t s h a r e s w i t h a l l Monte C a r l o c a l c u l a t i o n s the p r o b l e m of s t a t i s t i c a l e r r o r s and (sometimes) b i a s . In the s i m u l a t i o n s o f e x t e n s i v e systems, i n a d d i t i o n , there i s t h e a p p r o x i m a t i o n o f a u n i f o r m f l u i d by a f i n i t e p o r t i o n w i t h (say) p e r i o d i c boundary c o n d i t i o n s . The l a t t e r a p p r o x i m a t i o n appears t o be l e s s s e r i o u s i n quantum c a l c u l a t i o n s t h a n i n c o r r e s p o n d i n g c l a s s i c a l ones. We c a n g i v e h e r e o n l y a s k e t c h o f the b a s i s o f the method; f o r more d e t a i l s c o n s u l t r e f e r e n c e (£) and (£> • 3N I n d i m e n s i o n l e s s form and i n t h e space R f o r an N body s y s t e m , S c h r o d i n g e r ' s e q u a t i o n may be w r i t t e n as 2

-[V +V(R) ]*(R) = E<MR)

(12)

where V(R) i s t h e t o t a l p o t e n t i a l energy. Suppose t h e p o t e n t i a l energy i s bounded from below (13)

V(R) > - V . Q

Then Eq.

(12) may be r e w r i t t e n a s (-V +V(R)+V )iMR) = (E+V )iMR) 2

Q

0

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(14)

224

COMPUTER

Consider Green's which s a t i s f i e s

function

for

the

(-V +V(R)+V )G(R,R ) 2

0

Q

=

MODELING O F M A T T E R

operator

on the

6(R-R )

left,

( 1 5 )

Q

a n d a l l a p p r o p r i a t e b o u n d a r y c o n d i t i o n s f o r \p ( R ) . For example, i n s i m u l a t i n g an e x t e n s i v e system u s i n g a f i n i t e number o f p a r t i c l e s and p e r i o d i c b o u n d a r y c o n d i t i o n s , G must be p e r i o d i c w i t h r e s p e c t t o e v e r y c o ordinate. Equation 1 3 implies that G ( R , R Q ) i s always n o n n e g a t i v e and c a n be i n t e r p r e t e d as a d e n s i t y function. With the help o f such a Green's f u n c t i o n , E q . ( 1 4 ) may b e f o r m a l l y i n t e g r a t e d t o g i v e *(R)

=

(E+V )

G(R,R')iHR')dR'

Q

•

(16)

T h i s e q u a t i o n may b f u n c t i o n ip(n) t the n iterate. Asymptotically i p ( ) i s p r o p o r t i o n a l t o I ^ Q , t h e g r o u n d s t a t e wave function. The c o e f f i c i e n t is asymptotically constant i f E i n E q . (16) i s set equal to the ground s t a t e energy E Q . I t i s n o t d i f f i c u l t t o see t h a t i f a t any s t a g e n o n e h a s a p o p u l a t i o n o f p o i n t s {R][ ^ } d r a w n a t r a n d o m f r o m \p( ' , a n d i f o n e s a m p l e s {Rj[ +1) } a t t

a

n

n

n

n

n

random u s i n g t h e d e n s i t y f u n c t i o n ( E + V Q ) G ( R ^ ^ ) , R ^ ^ ) , t h e n t h e e x p e c t e d d e n s i t y o f t h e new p o p u l a t i o n n e a r R is 1

(E+V )ij; Q

( n )

(R')dR'

E i^

( n + 1 )

(R)

n

(17)

T h i s d e f i n e s one s t e p o f a random w a l k whose asympt o t i c d e n s i t y i s ijjg (R) . G ( R , R o ) i s n o t known e x p l i c i t l y (or by q u a d r a t u r e ) f o r any b u t t h e most s i m p l e (and u n i n t e r e s t i n g p r o b lems) . But i t i s c l e a r l y r e l a t e d to the s o l u t i o n of a d i f f u s i o n problem for a p a r t i c l e s t a r t i n g at R Q i n a 3 N d i m e n s i o n s p a c e and s u b j e c t t o a b s o r p t i o n p r o b a b i l i t y V(R) + V Q per u n i t time. We t h e r e f o r e expect t o be a b l e t o sample p o i n t s R f r o m G ( R , R Q ) conditional on R o . I t t u r n s o u t t o b e p o s s i b l e b y means o f a r e c u r s i v e random walk i n w h i c h e a c h s t e p i s drawn from a known G r e e n ' s f u n c t i o n f o r a s i m p l e s u b d o m a i n o f t h e f u l l space f o r the wavefunction. References (£) and (6) c o n t a i n a t h o r o u g h d i s c u s s i o n o f t h i s essential t e c h n i c a l p o i n t , and a l s o o f t h e methods w h i c h p e r m i t t h e a c c u r a t e c o m p u t a t i o n o f t h e e n e r g y and o t h e r q u a n tum e x p e c t a t i o n s s u c h a s t h e s t r u c t u r e f u n c t i o n , m o mentum d e n s i t y , B o s e - E i n s t e i n c o n d e n s a t e f r a c t i o n , a n d

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

17.

KALOS ET AL.

Quantum Liquids and

Crystals

225

crystal structure. One t e c h n i c a l p o i n t i s w e l l w o r t h r e i t e r a t i n g h e r e : i f o n e m o d i f i e s E g . (16) by m u l t i p l y i n g t h r o u g h by ^ T ( R ) , a t r i a l f u n c t i o n o f the k i n d used i n v a r i a t i o n a l methods, t h e n the Monte C a r l o v a r i a n c e o f a l l q u a n t i t i e s may b e c o n s i d e r a b l y r e d u c e d . When t h i s t r a n s f o r m a t i o n o f E q . (16) is carried out, i t i s p o s s i b l e t o e s t a b l i s h an e s t i m a t o r f o r the e n e r g y whose v a r i a n c e v a n i s h e s i n t h e l i m i t ~* ^0* I n p r a c t i c e we f i n d t h a t o p t i m u m v a r i a t i o n a l w a v e f u n c t i o n s always r e d u c e the v a r i a n c e o f the energy by large ratios. L i m i t e d e x p l o r a t i o n h a s a l s o shown that s i g n i f i c a n t departures from such t r i a l functions may c h a n g e t h e v a r i a n c e ( u s u a l l y , but not always, for the worse) but t h a t w i t h i n s t a t i s t i c s the answers agree. Results

Obtained wit

A number o f s y s t e m s h a v e b e e n s t u d i e d w i t h a l gorithms o f the c l a s s d e s c r i b e d here. A l l t h o s e t o be d i s c u s s e d here used the a c c e l e r a t i o n technique o u t l i n e d a t t h e end o f the p r e v i o u s s e c t i o n by e m p l o y i n g a ip w h i c h h a d t h e f o r m o f E q . (6) f o r f l u i d s a n d Eqs. ( 7 , 8) f o r crystals. C e p e r l e y , C h e s t e r , a n d K a l o s c a r r i e d o u t two studies (15_, 16_) o f b o s o n s y s t e m s i n w h i c h p a i r Y u k a wa forces (one t e r m w i t h e > 0 i n E q . ( 1 1 ) ) were used. F o r t h e most p a r t , t h e e n e r g y v a l u e s a g r e e d rather well with variational results, usually lying o n l y 1% l o w e r . I n c e r t a i n c a s e s d i s a g r e e m e n t s o f up t o 4% w e r e n o t e d . On t h e o t h e r h a n d , t h e r a d i a l d i s t r i b u t i o n f u n c t i o n was f o u n d t o b e s i g n i f i c a n t l y sharpe r i n t h e GFMC c a l c u l a t i o n s : t h e f i r s t p e a k o f the r a d i a l d i s t r i b u t i o n was u s u a l l y 10% m o r e p e a k e d t h a n i n the c o r r e s p o n d i n g v a r i a t i o n a l calculation. The g e n e r a l l y g o o d a g r e e m e n t w i t h t h e variational e n e r g i e s i n d i c a t e s t h a t the e q u a t i o n s o f s t a t e o f f l u i d s a n d c r y s t a l s w i t h Yukawa a n d s i m i l a r forces a r e g i v e n a d e q u a t e l y f o r most p u r p o s e s by the o r d i n a r y v a r i a t i o n a l method. F o r the s t r o n g l y c o u p l e d e l e c t r o n fermion f l u i d ( i . e . a t low d e n s i t i e s ) one c a n e s t i m a t e t h a t the change i n energy from the c o r r e s p o n d i n g Bose fluid is small. T h i s suggests t h a t the e r r o r i n u s i n g a p r o d u c t t r i a l f u n c t i o n a s i n E q . (9) i s c o r r e s p o n d ingly small. B u t t h e r e i s one p o i n t a b o u t t h e c o m p a r i s o n o f GFMC t o v a r i a t i o n a l r e s u l t s t h a t seems g e n e r a l l y a p p l i c a b l e and i m p o r t a n t : v a r i a t i o n a l r e s u l t s for the c r y s t a l e n e r g i e s are c l o s e r t o the e x a c t n u m e r i c a l r e s u l t s than are c o r r e s p o n d i n g r e s u l t s f o r the fluid. T h a t means t h a t t h e e s t i m a t i o n o f m e l t i n g a n d f r e e z i n g T

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

226

C O M P U T E R M O D E L I N G OF

MATTER

d e n s i t i e s from v a r i a t i o n a l r e s u l t s w i l l have a s y s t e ­ matic bias favoring c r y s t a l states. K a l o s , Levesque, and V e r l e t (6) r e p o r t e d a s t u d y on t h e hard-sphere f l u i d and c r y s t a l . They found t h e energy about 3-5% deeper t h a n had been o b t a i n e d v a r i a t i o n a l l y (17) and a s t r u c t u r e f u n c t i o n some 10-20% s h a r p e r . The a u t h o r s a l s o d e v e l o p e d a p e r t u r b a t i o n t h e o r y c o n n e c t i n g h a r d - s p h e r e and o t h e r s t r o n g l y r e ­ p u l s i v e p o t e n t i a l s . Using t h i s r e l a t i o n they estima­ t e d a minimum energy f o r f l u i d He-4 w i t h L e n n a r d Jones p o t e n t i a l s (Eq. (10)) as -6.8±0.2°K/atom o c c u r i n g a t 1.0±.l o f t h e e x p e r i m e n t a l l y o b s e r v e d d e n s i t y , P0C u r r e n t l y , W h i t l o c k e t a l . (18) have been t r e a t ­ i n g t h e Lennard-Jones f l u i d and FCC c r y s t a l by means o f t h e GFMC method. The a n a l y s i s and c a l c u l a t i o n o f certain correction l i m i n a r y r e s u l t s suppor sults. The e q u i l i b r i u m f l u i d i s found a t p/po = 1.03 w i t h an energy o f -6.85°K/atom. T h i s i s i n s t r i k i n g c o n t r a s t w i t h t h e v a r i a t i o n a l t r e a t m e n t based on a p r o d u c t w a v e f u n c t i o n , Eq. (6) f o r w h i c h no c a l c u l a t i o n has g i v e n a r e s u l t deeper t h a n -6°K. The e x p e r i m e n t a l r e s u l t i s -7.14°K. Thus t h e L e n n a r d - J o n e s p a r a m e t e r s o f Eq. (10) g i v e s u b s t a n t i a l l y b e t t e r e q u a t i o n o f s t a t e t h a n had been supposed on the b a s i s o f v a r i a ­ tional calculations. Variational calculations with t h e p r o d u c t t r i a l f u n c t i o n g i v e r a t h e r c r u d e upper bounds t o the e q u a t i o n o f s t a t e and hence r a t h e r l i m i t e d i n f o r m a t i o n about t h e He-He p o t e n t i a l . Two t h e o r e t i c a l s t u d i e s have shed l i g h t on t h e d i s c r e p a n c y between the v a r i a t i o n a l and GFMC e n e r g i e s f o r l i q u i d He-4. Chang and C a m p b e l l (.19) e s t i m a t e d by a p e r t u r b a t i o n a l t h e o r y t h a t about h a l f t h e d i s c r e p a n ­ cy between the two r e s u l t s c o u l d be a s c r i b e d t o t h e n e g l e c t o f three-body c o r r e l a t i o n s i n t h e t r i a l f u n c ­ t i o n . More r e c e n t l y , P a n d h a r i p a n d e (20) used a t r i a l f u n c t i o n w i t h a three-body c o r r e l a t i o n corresponding t o a l i n e a r i z e d " b a c k - f l o w " (21). W i t h i n t h e frame­ work o f i n t e g r a l e q u a t i o n s o f t h e HNC t y p e , he c a l c u ­ l a t e d a minimum energy o f (-6.72±0.2)°K a t a P / P Q o f 1.05. These t h r e e - b o d y c o r r e l a t i o n e f f e c t s are un­ d o u b t e d l y i m p o r t a n t i n He-3 as w e l l . A s i g n i f i c a n t p a r t o f t h e d i s c r e p a n c y between t h e s t r u c t u r e f u n c t i o n deduced from v a r i a t i o n a l c a l c u l a ­ t i o n s and from e x p e r i m e n t can a l s o be a s c r i b e d t o t h e n e g l e c t o f three-body c o r r e l a t i o n s i n the t r i a l f u n c ­ tions. F i g u r e 1 shows a c o m p a r i s o n o f e x p e r i m e n t a l , v a r i a t i o n a l , and GFMC e s t i m a t e s o f S ( k ) . I t i s clear t h a t t h e l a t t e r agrees b e t t e r w i t h e x p e r i m e n t t h a n t h e

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

Figure 1. Comparison of structure functions for He-4 at equilibrium density. The solid line shows the smoothed experimental data of Achter and Meyers (Phys. Rev. (1969) 188, 291) with bars indicating one standard deviation. S(K) computed variationally is shown by triangles; the circles show the results computed using GFMC.

-a

to to

C O M P U T E R M O D E L I N G OF

228

MATTER

variational result. I t i s a l s o c l e a r , i n s p i t e o f the improvement o f e q u a t i o n o f s t a t e and o f S(k) t h a t r e s u l t s from the use o f GFMC, t h a t the L e n n a r d - J o n e s p o t e n t i a l c a n n o t be the c o r r e c t one. Three body p o t e n t i a l e f f e c t s a r e s m a l l i n He-4; we e s t i m a t e t h a t a t p = P Q the t r i p l e d i p o l e f o r c e c o n t r i b u t e s about 0.16°K/atom, f a i r l y c l o s e t o the r e s u l t g i v e n by Murphy and B a r k e r (22) . There i s good r e a s o n t o b e l i e v e i n g e n e r a l t h a t t h e e f f e c t o f t h r e e body f o r c e s on t h e e q u a t i o n of s t a t e w i l l be s m a l l as s u g g e s t e d by t h i s p a r t i c u l a r r e s u l t . T h i s i s a consequence o f the h i g h energy o f t h e f i r s t e x c i t e d s t a t e of helium. I t i s not u n r e a s o n a b l e t o hope t h a t two body f o r c e s d e t e r m i n e d from s c a t t e r i n g , v i r i a l and t r a n s p o r t d a t a s h o u l d a l s o be c o n s i s t e n t w i t h the p r o p e r t i e s o f h e l i u m l i q u i d s and c r y s t a l s . Conclusions I n t h i s b r i e f r e v i e w we have chosen t o concen­ t r a t e upon the c h a r a c t e r o f some new methods f o r t h e Monte C a r l o m o d e l l i n g o f quantum systems. I n so d o i n g we have emphasized c e r t a i n d e f i c i e n c i e s o f the o l d e r method w h i c h r e s t s upon t h e p r o d u c t t r i a l f u n c t i o n i n a v a r i a t i o n a l expression. I t i s n e c e s s a r y t o remark t h a t t h i s l a t t e r t e c h n i q u e remains u s e f u l : i t i s a r e a s o n a b l e g u i d e t o t h e phenomena i n quantum systems and f o r s o f t - c o r e systems g i v e s r e s u l t s f o r the equa­ t i o n o f s t a t e o f l i q u i d s and c r y s t a l s w h i c h are ade­ quate f o r most p u r p o s e s . The e x t e n s i o n o f the Monte C a r l o v a r i a t i o n a l method t o i n c l u d e t h r e e - b o d y c o r r e ­ l a t i o n s i s s t r a i g h t f o r w a r d but c o m p u t a t i o n a l l y s l o w ; i t s h o u l d be done t o p r o v i d e r e l i a b l e c h e c k s on t h e t h e o r e t i c a l work on s u c h e f f e c t s i n He-4. The s t u d y o f inhomogeneous systems and m i x t u r e s remains l a r g e l y unexplored. The f e r m i o n and Green's F u n c t i o n Monte C a r l o a r e i m p o r t a n t i n t h e m s e l v e s and i n t e r e s t i n g as h i n t s t o t h e r i c h n e s s o f methodology t h a t can be b r o u g h t t o b e a r on the c o m p u t a t i o n o f quantum systems. I n t h e s h o r t t e r m we e x p e c t t o broaden the s p e c i f i c t r i a l f u n c t i o n s used i n f e r m i o n Monte C a r l o t o p e r m i t the more a c c u r a t e s t u d y o f He-3 and t h e t r e a t m e n t o f more r e a l i s t i c models o f n u c l e a r and n e u t r o n m a t t e r . We e x p e c t a l s o t o t r y a v a r i a n t i n Quantum C h e m i s t r y problems. The most immediate r e s e a r c h we p l a n w i t h the Green's F u n c t i o n Monte C a r l o i s t h e e x p l o r a t i o n o f the e q u a t i o n of s t a t e of He-4 l i q u i d s and c r y s t a l s w i t h two body p o t e n t i a l s w h i c h f i t more d a t a t h a n the

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

17.

KALOS ET AL.

Quantum Liquids and Crystals

229

Lennard-Jones-de B o e r - M i c h e l s form we have used. The e x p l o r a t i o n o f inhomogeneous systems i s a l o g i c a l n e x t s t e p . A t t h e moment o n l y GFMC o f f e r s an unambiguous approach t o t h e c a l c u l a t i o n o f t h e d e n s i t y p r o f i l e a t an i n t e r f a c e , f o r example. F o r t h e f u t u r e we a n t i c i p a t e t h e development o f p r a c t i c a l methods f o r t h e t r e a t m e n t o f systems a t f i n i t e t e m p e r a t u r e s (some o f t h e t e c h n i c a l problems o f e x t e n d i n g GFMC t o t e m p e r a t u r e s g r e a t e r t h a n z e r o have r e c e n t l y been s o l v e d f o r t h e two-body h a r d sphere p r o b l e m (23)) . An e x t e n s i o n t o p e r m i t t h e e x a c t o r v e r y r e l i a b l e t r e a t m e n t o f f e r m i o n systems seems p o s s i b l e and u s e f u l . Acknowledgment T h i s work was s u p p o r t e o f Energy, C o n t r a c p a r t by t h e N a t i o n a l S c i e n c e F o u n d a t i o n under G r a n t DMR-74-23494 t h r o u g h t h e M a t e r i a l S c i e n c e C e n t e r , Cornell University. Literature Cited 1 2 3 Teller, 4 Statistical 5 6 7 8 9 10 11 12 13 14 15

P a n d h a r i p a n d e , V.R. and B e t h e , H.A., Phys. Rev. C, (1973), 7, 1312. M c M i l l a n , W.L., Phys. Rev., (1965), 138 A, 442. M e t r o p o l i s , N., R o s e n b l u t h , A.W., R o s e n b l u t h , M.N., A.H., and Teller, E., J. Chem. Phys., (1953) 2 1 , 1087. C e p e r l e y , D.M. and Kalos, M.H., "Quantum ManyBody P r o b l e m s " , Chap. IV o f Monte Carlo Methods in Physics, K. B i n d e r , e d . ; in press, Springer-Verlag. C e p e r l e y , D.M., Chester, G.V., and Kalos, M.H., Phys. Rev. B, (1977), 16, 3081. K a l o s , M.H., L e v e s q u e , D. and Verlet, L., Phys. Rev. A., (1974), 9, 2178. Wu, F.Y. and Feenberg, E., Phys. Rev., (1962), 128, 943. S c h i f f , D. and Verlet, L., Phys. Rev., (1967), 160, 208. Z a b o l i t z k y , J., Phys. Rev. A., ( 1 9 7 7 ) , 16, 1258. Day, B., Rev. Mod. Phys., in press. C e p e r l e y , D.M., to be published. S t e v e n s , F.A. and P o k r a n t , M.A., Phys. Rev. A, (1973), 8, 990. Freeman, D.L., Phys. Rev. B, (1977), 15, 5513. C a r r , W.J., Phys. Rev., (1961), 122, 1437. C e p e r l e y , D.M., C h e s t e r , G.V. and Kalos, M.H.,

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16 17 18 19 20 print, 21 22 23

Phys. Rev. D, in press. C e p e r l e y , D.M., C h e s t e r , G.V. and Kalos, M.H., Phys. Rev. D, (1976), 13, 3208. Hansen, J.-P., L e v e s q u e , D. and Schiff, D., Phys. Rev. A, (1971), 3, 776. W h i t l o c k , P.A., K a l o s , M.H., C h e s t e r , G.V. and C e p e r l e y , D.M., to be published. Chang, C.C. and C a m p b e l l , C.E., Phys. Rev. B, (1977), 15, 4238. Pandharipande,V.R., University of Illinois pre­ December, (1977), I L L - ( T H ) - 7 7 - 4 1 . Feynman, R.P. and Cohen, M., Phys. Rev., (1956), 102, 1189. Murphy, R.D. and B a r k e r , J.A., Phys. Rev. A, (1971), 3, 1037. Whitlock, P.A. and K a l o s , M.H., s u b m i t t e d t o J. Comp. P h y s . , (1978)

RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

18 From Microphysics to Macrochemistry via Discrete Simulations JACK S. TURNER Center for Statistical Mechanics and Thermodynamics, University of Texas, Austin, TX 78712 Historically there are two distinct classes of problems in chemistry to which discret applied widely and with considerable success: At one extreme the bulk physical properties of atomic and molecular fluids are studied as the "exact" dynamical evolution of a collection of representative particles is followed in "computer experiments" using the well-established method of molecular dynamics (1-10). At the other extreme similar techniques (11-16) are used to explore the chemical transformations which may occur in an isolated collision between potentially reactive species (e.g., in a very dilute gas). Between these mutually exclusive limits lies an increasingly important area of chemistry in which macroscopic properties are a direct consequence of cooperative interplay between molecular motion and chemical dynamics. Of immediate interest in this area are chemical systems which may undergo nonequilibrium i n s t a b i l i t i e s and transitions between different regimes of macroscopic physicochemical behavior (See Ref. 17 for a recent survey of the f i e l d ) . Fully analogous to equilibrium phase transitions and c r i t i c a l phenomena, these "nonequilibrium phase transitions" present even more formidable d i f f i c u l t i e s in both microscopic theory and laboratory experiment. Hence an obvious need exists for detailed computer simulations to provide "experimental" data for theorists and "theoretical" guidance for experimentalists. Since the development of appropriate simulation methods in this area is in its infancy, the purpose of this paper will be to provide motivation for such investigations in the more familiar context of molecular dynamics in classical f l u i d s , to review feasible methods at two levels of description which have emerged in the last few years (18, 19, 20, 21, 22), and to i l l u s trate their application using two simple models which exhibit chemical i n s t a b i l i t i e s and transitions. In a l l three classes of problems the systems of interest are characterized by the interaction of large numbers of individual degrees of freedom. It is this feature which leads to great theoretical d i f f i c u l t i e s in both classical and quantum systems, 0-8412-0463-2/78/47-086-231$08.25/0 © 1978 American Chemical Society In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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and hence provides strong motivation for numerical "experiments" using high-speed digital computers. Appropriate simulations i n volve molecular dynamics [MD] (1-1J3) or Monte Carlo [MC] [7_ 23, 24, 25) methods which require the details of microscopic interactions and processes. During the course of a simulation this essential information is obtained from a potential energy function (or surface), and i t is our limited a b i l i t y to provide this i n formation which serves to delineate the two extreme problem classes for practical application. In the f i r s t problem class mentioned above (hereinafter called class A), a collection of particles (atoms and/or molecules) is taken to represent a small region of a macroscopic system. In the MD approach, the computer simulation of a laboratory experiment is performed in which the "exact" dynamics of the system is followed as the particles interact according to the laws of classical mechanics. Used extensively to study the bulk physical properties of classical f l u i d s , such MD simulations can yield information about equilibrium (See Ref. ^ for a review) in addition to the equation of state and other properties of the system at thermodynamic equilibrium (7,, 8_ for example). Current a c t i v i t i e s in this class of microscopic simulations is well documented in the program of this Symposium. Indeed, the state-of-the-art in theoretical model-building, algorithm development, and computer hardware is reflected in applications to relatively complex systems of atomic, molecular, and even macromolecular constituents. From the practical point of view, simulations of this type are limited to small numbers of particles (hundreds or thousands) with not-toocomplicated inter-particle force laws (spherical symmetry and pairwise additivity are typically invoked) for short times (of order 10"' to 10"^ second in liquids and dense gases). In direct contrast to simulations of physical properties dominated by inter-particle forces, the second problem class (class B) concerns interactions among intramolecular and intermolecular degrees of freedom in an event which produces a chemical change in the participants. Severely limited by the need for an accurate quantum mechanical potential energy surface, simulations in this area typically involve classical trajectory ( i . e . , MD) calculations for the simplest chemical reactions 9

2

(11-16).

It is not surprising that the two main classes of microscopic simulations have evolved quite independently. Aside from the obvious problem of calculating potential energy functions (surfaces), the greatest computational d i f f i c u l t y arises in treating systems with multiple time scales. Dynamical simulations within class A are feasible because the bulk properties of interest can be determined on a time scale corresponding to a computationally f i n i t e number of molecular c o l l i s i o n s . When the most important events are rare on this time scale, one rapidly reaches the limits of f e a s i b i l i t y for detailed molecular dynamics

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simulations. With rare exceptions, collisions in which the p a r t i cipants are chemically transformed are indeed rare dynamical events. Hence the combination of potential energy surface and time-scale d i f f i c u l t i e s means in particular that complex chemical reactions in a many body system are beyond the reach of truly microscopic simulations. That i s , the systems of immediate interest in the intermediate class (class C) cannot be treated by straightforward extension (and combination) of class A and class B methods. This conclusion is certainly nothing new to workers in either of the two main areas of simulation. Indeed much effort in recent years has been directed at precisely the problem which is addressed here in the specific context of chemical i n s t a b i l i ties and nonequilibrium phase transitions: How can we simulate the evolution and steady state behavior of systems in which the events of greatest interest are rare on the time scale of i n dividual c o l l i s i o n s , bu collective many-body dynamics Multiple Time Scales, Bottlenecks and Many-Body Dynamics Systems in which problems of multiple time scales and of "bottlenecks in phase space" occur are hardly exceptional, and promising methods of theory and discrete-event simulation have been developed recently in several important areas. In a marriage of molecular dynamics (and Monte Carlo methods) to transition state theory (2&), for example, Bennett (2Z) has developed a general simulation method for treating a r b i t r a r i l y infrequent dynamical events ( e . g . , an enzyme-catalyzed reaction process). An entirely different class of problems has been the recent focus of Adelman, Doll, and coworkers (28, 29, 30, 31_, 32J, who have investigated the microscopic dynamics of scattering, sorption, and reaction at the gas-surface interface. By taking the manybody l a t t i c e dynamics into account in a novel theoretical model based on nonequilibrium s t a t i s t i c a l mechanics, these authors have developed a new method of molecular dynamics involving generalized Langevin equations of motion. The new simulation techniques of Bennett (27.) and Adelman and Doll (28_, 29) i l l u s t r a t e well several important aspects of the present problem. The former considers relatively fast but infrequent events, includes a l l relevant degrees of freedom expressible in a potential energy surface, and concentrates on the dynamical bottlenecks separating regions of the system's state (phase) space (e.g., products from reactants). In contrast, the latter method treats only a subset of degrees of freedom in detail (e.g., the colliding atom and those surface atoms and neighbors most directly involved), while including in an average way the effect of coupling to other degrees of freedom of the bulk l a t t i c e . It is important to realize that both methods treat individual events or sequences of events in great d e t a i l , yielding s t a t i s t i -

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cal information about a single overall process. Like the class B chemical simulations mentioned above, these methods serve to characterize the "elementary processes" (that i s , the reactive events) with which the problems of class C are concerned. In other words, the "fundamental units" of interest in the intermediate class of systems are not simply the atoms or molecules themselves, but rather the reactive collisions and transformations. As we shall see, i t is the space-time dynamics and interactions of these new "units" (mediated, of course, by other degrees of freedom via elastic and inelastic c o l l i s i o n s , intramolecular dynamics . . . ) which determine ultimately the macroscopic behavior of such systems. In this respect we see more clearly the connection to the problems of class A. At e q u i l i brium phase transitions in classical f l u i d s , for example, the important "units" are the clusters or nuclei of the new phase embedded in the i n i t i a l phase. These units are strongly coupled to their surroundings (e.g., via c o l l i s i o n s ) , and in addition are characterized by their ow The time-scale characteristic of size and/or composition changes in individual nuclei is obviously much longer than that of the detailed microscopic dynamics. In particular, the timedelay associated with spontaneous development of a "critical nucleus" has been a source of great computational d i f f i c u l t y in simulations of both 1iquid+solid (33) and vapor+1 iquid (34-42) phase transitions in simple classical fluids. Before turning to an analogous situation at the "nonequilibrium phase transitions" which are of immediate interest, therefore, let us f i r s t illustrate the key problems and innovations in simulation which are already evident in MD studies of low-order clustering and homogeneous nucleation in the vapor phase (34-42). Molecular Dynamics in Dilute Gases In MD simulations of dense f l u i d s , the classical many-body equations of motion can be integrated e f f i c i e n t l y using a singletime-step method (e.g., Refs. 3^ and 4J. This is so because at any instant of time a l l particles are interacting strongly with a number of others. In a dilute gas, however, only a small fraction of the particles are strongly interacting at any time. If a Lennard-Jones 6-12 (LJ) potential is assumed for a model of f l u i d Argon, for example, most atoms are at most very weakly interacting via the long-range attractive t a i l of the potential. With a timestep selected for the few colliding pairs of atoms, the efficiency of the usual MD methods is lost. Recognizing that each atom spent most of its time in essentially free flight between binary c o l l i sions, Harrison and Schieve (34_, 36J introduced a cutoff ( R „ ) in the range of the LJ potential and combined the Alder ( 1 ) freeflight method for "isolated" atoms with a Rahman (3_) or Verlet (4) continuous-potential method for the remaining atoms. (For details of the dilute gas algorithm see Ref. 36_.) The result was

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a hybrid MD algorithm with which simulations of a dilute c l a s s i cal gas could be performed e f f i c i e n t l y in two and three dimensions (34, 35, 36, 37, 38). Low-Order Clustering in a Computer Model of Argon Vapor. Using the algorithm which decoupled isolated atomic motion from collisions between atom pairs, Harrison and Schieve (34, 35j f i r s t studied the approach to equilibrium in a 2D gas of (LJj Argon. At lower temperatures and higher pressures, low-order clustering was observed, beginning with the formation of bound states (dimers) by the three-body channel (35, 36^, 37, 38) A + A J A* 2

A* + A + A 2

2

+A

(1)

In the f i r s t step an unstabl the dimer is formed whic energy is then removed by a third atom in the second step to form the stable dimer A . A typical sequence is shown in Figure 1, in which a dimer 12 is created in a c o l l i s i o n with 3 and destroyed after a few vibrations in a c o l l i s i o n with 6. Detailed MD studies of the dimer formation "reaction" have been carried out in two and three dimensions (_35, 36, ^37, 38). The result of greatest interest for the present study is the time-scale problem i n troduced by the occurrence of the "chemical reaction" (Eq. 1). Since the dimer-formation process is a dynamically rare event, lengthy computer runs (of order 40,000 collisions in 2.4 x 10"' sec. model time) were required to obtain s t a t i s t i c a l l y reliable values for the dimer formation rates, lifetimes, and other properties. The problem to be faced in this situation is a classic one in computer simulation: What level of microscopic detail must be included in order to retain the essential features of the system at a not unreasonable cost? The solution of this problem in the dimer study illustrates well the general approach of this paper: Beginning with the most detailed simulation possible (which then serves as a benchmark), develop a hierarchy of simulations in which the computer model at each level incorporates, in an average way, at least, those features of lower levels relevant to the questions of interest. In this way the qualitative behavior and properties of a system can be mapped with great efficiency, and areas of particular interest so identified could be subjected to more careful, even quantitative, study in lowerlevel simulations. For the dimer simulation two simplifications were made: First was the obvious step of restricting the study to two dimensions, at least i n i t i a l l y . Second, the continuous LJ potential was replaced by a square-well hard-core (SWHC) potential with parameters determined empirically from equation-of-state data for Argon (cf, Ref. _43, p. 158). With these two refinements i t 2

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was possible not only to improve the s t a t i s t i c s of the dimer c a l culations, but also to consider s t i l l lower temperatures and higher pressures at which trimers, tetramers and other low-order clusters could be formed (39, 40., 41). Molecular Dynamics Evidence for Vapor-Liquid Nucleation. Specializing the MD model now to a 2D system of 100 SWHC "Argon" atoms, let us review briefly relevant aspects of these simulations. The SWHC potential is defined with well-depth e = 2.305 x 10-14 erg, hard core radius o\ = 2.98 x 10" cm, and range of attractive interaction a = 1.96 o\. The atomic mass is m = 6.628 x 10" g. With periodic boundary conditions, the system is adiabatic, at constant volume, total energy, and number of atoms. I n i t i a l l y the atoms are displaced slightly from the sites of a square lattice 3.334 x 10" cm (112-08 a ) on a side, and are given a fixed speed with random velocity orientation. This fixes the total energy of the ( i n i t i a l l y non-interacting) system to be the i n i t i a are formed (with negative potential energy), therefore, the kinetic energy ( i . e . , temperature) increases accordingly until an equilibrium distribution of monomers, dimers, trimers, . . . is eventually reached. A quite remarkable feature of the MD studies of Zurek and Schieve (40, 41, 42) using the above computer model is the direct observation oT~nucTeation in the vapor-liquid phase transition region. The phase transition i t s e l f is clearly indicated in Figure 2; the heat capacity C = 3E(T*)/8T* is seen to increase dramatically near Tgq * 0.5. Here T* = kT/e and the total energy E* = E/e (or Pfn) is plotted against the equilibrium kinetic temperature Tj£q. (Initial conditions with negative total energies are obtained by decreasing the temperature in the final state of higher-energy runs.) In Figure 3 a "supercritical" cluster is clearly evident in a "snapshot" of the system at Tf =0.2 (41J. This qualitative indication that nucleation has occurred is made quantitative in terms of the physical cluster distributions plotted in Figure 4. Here a dramatic change in the distribution is seen as the phase transition region is entered in 4c and4d. The gap which appears separates clusters into sub- and superc r i t i c a l classes, and the width of the gap provides bounds on the number of atoms in the "critical" cluster. It is important to realize that nucleation has introduced yet another time scale into the MD simulation--the time required for spontaneous generation of the "critical" nucleus which, in a non-adiabatic system, initiates an irreversible transition from the metastable i n i t i a l vapor state to the stable liquid phase. Although the implications of this time scale have not been treated in the Argon studies, they will be examined in the simulations of nonequilibrium phase transitions which follow. (Note: These nucleation results have now been verified in a recent MD/MC study by Rao, Berne, and Kalos, Ref. 54.) 8

2

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Journal of Chemical Physics

Figure 1. Plot of particle trajectories involved in the formation and destruction of a dimer (12) during a molecular dynamics simulation of 2-D (LJ) Argon (35)

7

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Journal of Chemical Physics

Figure 2. The spatial distribution of the particles in the system at a given instant shows a well-developed droplet containing 19 particles. The size of the points in the picture corresponds to the size of the outer wall of the potential (a ). T*ISIT 2

ss -0.2, T*j5 s£ 0.5 (41). Q

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1.2

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Figure 3. The E(T*) [T* dependence, showing the characteristic shape for T* ^ 0.5. T* has already evolved in the higher (T* i 0) temperature for some time. The slope is the specific heat c . After Ref. 42. IXIT

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Chemical Instabilities, Nonequilibrium Phase Transitions, and Dissipative Structures Let us now digress briefly from matters of simulation to i n troduce the kinds of chemical systems in which nonequilibrium phase transitions can occur. This is most conveniently done at the macroscopic level of deterministic chemical kinetics. Consider a chemical system under open or closed conditions so that i t is possible to maintain a time-independent macroscopic state. At or near thermodynamic equilibrium, the macroscopic description of such a system yields a unique stationary solution that is always stable with respect to arbitrary perturbations. S u f f i ciently far from equilibrium, however, the governing kinetic equations for nonlinear chemical systems may admit solutions other than the continuous extension of the equilibrium solution into the nonequilibrium domain. Beyond a c r i t i c a l distance from e q u i l i brium, moreover, this thermodynamic branch may become unstable in response to fluctuation point of i n s t a b i l i t y , the response of the system to an i n f i n i t e s i mal disturbance leads ultimately to a new operating regime characterized by organization in space, time, or function. Whatever the outcome of such a macroscopic i n s t a b i l i t y , therefore, the i n stability i t s e l f originated in the response of the system to a fluctuation. Consequently a purely deterministic description of the system in terms of mean values alone is no longer adequate, and i t is essential to supplement the "average" description with a theory of fluctuations extended to nonlinear systems under farfrom-equilibrium conditions. At thermodynamic equilibrium fluctuations constitute a negligible correction to the macroscopic description of matter except near instability points such as phase transitions or c r i t i c a l points. Similarly, in nonequilibrium situations as well one expects that fluctuations become important only near points of nonequilibrium instability (see Ref. ]_7. for a detailed discussion). For nonlinear systems in which such i n s t a b i l i t i e s and transitions can occur, the role of fluctuations is even more dramatic than at equilibrium, due to the existence in general of many distinct "phases" compatible with the external conditions imposed on the system. This fact has been known for several years now, since the f i r s t (numerical) demonstration on simple chemical models of multiple solutions to the governing macroscopic differential equations (44_, 45_, _46L, 47). The variety of possible solutions includes tirne-independent homogeneous and inhomogeneous states as well as states which are organized in time (homogeneous chemical oscillations) or in time and space (travelling chemical waves). Which of these "dissipative structures" (V7) will arise under given conditions depends crucially on the specific type of perturbation to which the original state is i n i t i a l l y unstable. In terms of spontaneous fluctuations, which are present in general over a range of wavelengths, both the final state and the evolu-

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tion toward that state will be determined by the space-time characteristics of the fastest-growing unstable mode. As one would expect, developments in the theory of such phenomena have employed chemical models chosen rorc for analytical simplicity than for any connection to actual chemical reactions. Due to the mechanistic complexity of even the simplest laboratory systems of interest in this study, moreover, application of even approximate methods to more r e a l i s t i c situations is a formidable task. At the same time a detailed microscopic approach to any of the simple chemical models, in terms of nonequilibrium s t a t i s t i cal mechanics, for example, is also not feasible. As is well known, the method of molecular dynamics discussed in detail a l ready had its origin in a similar situation in the study of c l a s s i cal fluids. Quite recently, the basic MD computer model has been modified to include inelastic or reactive scattering as well as the elastic processes of interest at equilibrium phase transitions (18), and several applications of this "reactive" molecular dynamics~lRMD) method t cal i n s t a b i l i t i e s have been reported (1_8> 1_9, 22_). A variation of the RMD method will be discussed here in an application to a first-order chemical phase transition with many features analogous to those of the vapor-liquid transition treated earlier. Reactive molecular dynamics may be viewed as a microscopic computer experiment inasmuch as a l l relevant particle processes can be included to some degree. In particular the method treats elastic as well as inelastic or reactive collisions between p a r t i cles and, indeed, requires that the latter be relatively rare events in order to simulation actual chemical kinetics in a r e a l i s t i c way. When large enough systems are considered (e.g., thousands of particles) i t should be possible to "measure" temporal and spatial correlation functions, for example, and to make quantitative the notion of c r i t i c a l size and amplitude of fluctuations necessary to nucleate a macroscopic transition. A f i r s t step in the latter direction will be reported here. Applications to date of reactive molecular dynamics methods demonstrate f e a s i b i l i t y for the study of hundreds or thousands of particles involved in chemical reactions for which reactive proba b i l i t i e s do not vary too widely. The latter condition is essent i a l i f s t a t i s t i c a l l y significant numbers of a l l possible events are to occur within a practical computation period. Even i f the requirement for a large excess of elastic collisions is relaxed, however, reaction rates typical of experimental chemical systems demand simulation run times which approach the feasible limit except for quite small numbers of particles. Turning therefore to a higher level method for this type of system, one may treat numbers of particles in a small cell or volume element rather than individual particles, and invoke a Monte Carlo procedure for determining which reactive event will occur, how much time will elapse between events, and in which "cell" of the system the

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process will occur. Transport between adjacent cells is handled in a similar fashion. In such a method, only the events of greatest interest (reaction or transport) are treated, so that widely different rate constants need not be a problem. In the change from reactive molecular dynamics to stochastic simulation, much larger systems may be considered, and computational efficiency is gained at the expense of molecular detail. In addition, problems related to numerical "stiffness" associated with the deterministic d i f f e r ential equations (48, 49J are completely avoided in a stochastic simulation. Finally, a number of implementations of such techniques have been presented which give qualitatively reasonable results. Of particular significance is a recent formulation due to Gillespie (20J, in which the transition probabilities and Monte Carlo implementation are rigorous consequences of the very assumptions from which chemical master equations of the birth-anddeath type are derived. This fact is of great importance for the development and testin periments based on the latter formulation simulate directly a stochastic chemical evolution satisfying a master equation of the type which is central to theoretical developments in this area (17). A report of i n i t i a l investigations of chemical i n s t a b i l i ties using stochastic simulation methods will also be presented. In order to fix the main ideas most e f f i c i e n t l y , the microscopic chemical simulations will be presented in applications to simple chemical models which together exhibit most of the known types of i n s t a b i l i t i e s and transitions. As will be seen, both RMD and stochastic simulations treat the elementary c o l l i s i o n and transport processes which occur in a chemical mixture instead of the concentration (or populations) of the constituents themselves. In the language of a Markovian stochastic description of chemical kinetics (cf, Ref. 17), both of these methods may be expressed in terms of the probabTTity P ( T , y ) 6 x that the next event will occur in the infinitesimal time increment 6x following an interval T and will be an event of type y. Such a probability leads naturally to a procedure for simulating a system's time-evolution in a "computer experiment." Each method is distinguished therefore by an algorithm for evaluating P ( x , y ) and for implementing an i t e r ative procedure to simulate an evolution of a chemical system. Microscopic Chemical Simulations:

Reactive Molecular Dynamics

In designing a simulation or "computer experiment" in a particular context, one must f i r s t determine the level of detail needed to describe the phenomena of interest. For chemical reactions, in which the fundamental interactions are between atoms and molecules, one would expect to begin at the level of microscopic dynamics. Simulations of this kind follow the details of classical many-body dynamics derived from assumed intermolecular interactions, and constitute the by now familiar method of

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molecular dynamics. As we have seen, such microscopic simulations of chemical reactions are impractical at present, largely due to the enormous number of "uninteresting" dynamical events which must be followed in detail between the reactive collisions of greatest concern. One way of avoiding part of the d i f f i c u l t y is to adopt a molecular "pseudodynamics" approach in which molecules travel in free f l i g h t between instantaneous hard-sphere collisions at which reactive processes may occur. Such methods have been employed recently by Portnow (18), by Ortoleva and Yip (19), and by the present author (22) and are applied here to some questions raised in the deterministic and stochastic formulations of chemical i n s t a b i l i t i e s . First Order Chemical Phase Transition in a Cooperative Isomerization Reaction. A convenient model of a f i r s t order transition is provided by a reversible isomerization reaction in a macroscopically homogeneous system (2)

B

having activation energies e for the forward reaction and n/N per molecule of B present in the system for the reverse reaction, where N is the total number of molecules. The activation energy for the reverse reaction depends on the number Ng of B molecules present, and therefore introduces a cooperativity into the mechanism. In physical terms, this cooperativity expresses the stabilizing influence on a B isomer of intermolecular interactions with other isomers of the same type. Aside from being a convenient way to introduce nonlinearity into the simplest chemical model, this procedure yields a model which is isomorphic to a Bragg-Williams (mean field) model for monomolecular adsorption on a uniform surface. The latter model of the equilibrium twodimensional l a t t i c e gas is known to exhibit a first-order phase transition (cf, Ref. _50) and provides a convenient vehicle for an especially intuitive discussion of the stochastic theory of metastability and nucleation at chemical first-order transitions (22, 51). (The use here of this equilibrium example is motivated by practical considerations of theory and simulation, and does not limit the generality of the results and conclusions.) If the cooperativity is introduced into the rate constant for the reverse reaction via a "mean-field" dependence on Ng, then the equilibrium constant for the isomerization reaction takes the form K = exp (-e + nN /N) B

(3)

where the activation energies are expressed in units of kgT, with kg the Boltzmann constant and the T the absolute temperature. If N is fixed (closed system), then the kinetics of the reaction is given by a single macroscopic rate equation

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Microphysics to Macrochemistry via Discrete Simulations

W- - 0< k

N

- B> " N

e

" 0 B "

£

k

N

e

<>

n N B / N

4

Defining a new time variable T = k e" t and the mole fraction B - B / N gives 0

X

=

X

e

N

*1 = i -

x

- xe " e

(5)

nx

A linear stability analysis of the steady state XQ of this system implies stability whenever the real normal mode frequency u is negative, and instability otherwise, where xu> = -1 + nx - nx . Points of marginal s t a b i l i t y correspond to a> = 0 and, according to the quadratic expression for a), there may be 0, 1, or 2 points. It is easy to see that there is no instability for n < n 4. For n > nc» however, there is a region of e in which three steady states are possible. Typica Figure 5. Here the curve passing through the c r i t i c a l point (nc»£c>xc) (4,2,1/2) separates regions of 1 and 3 steady states. In the three state region, the upper and lower branches are stable, the middle branch unstable, according to linear stability analysis. If i t is supposed for convenience that e is an externally variable parameter of the model, then for n > n the transition between stable branches of the deterministic solution should occur discontinuously at the points of marginal stability (arrows, Figure 5). The similarity between this picture and the pressurevolume diagram for the equilibrium liquid-vapor transition is apparent. Indeed, according to the macroscopic van der Waals model of a nonideal gas (50) the liquid-vapor phase change in a homogeneous f l u i d should occur at a lower pressure than the reverse transition. That a single equilibrium transition pressure (e.g., dashed l i n e , Figure 5, an "equal-areas" construction) is observed regardless of the direction of change is a direct consequence of the response to spatially localized, finite-amplitude fluctuations in the i n i t i a l f l u i d state. Moreover, when such fluctuations are of insufficient size or amplitude to provide a stable nucleus for the transition, then states beyond the e q u i l i brium transition point may be realized. Such states are termed metastable, since they are stable only until a large enough fluctuation appears to form a "supercritical" nucleus. At which point along the metastable branch will destabilizing fluctuations appear, and how long will the metastable state remain "stable" prior to occurrence of c r i t i c a l fluctuations? Clearly such questions are beyond the scope of the deterministic description, and will be addressed here from the viewpoint of RMD simulation. 2

c

=

=

c

Reactive Molecular Dynamics. A microscopic computer model of a chemically reacting mixture is constructed with the following

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essential features: (1) Hard-sphere dynamics. The key assumption is that the most important chemical effects occur upon energy and momentum transfer at molecular c o l l i s i o n s , the net effect of attractive intermolecular forces being to alter the collision crosssection. Hpnce the microscopic molecular dynamics is simplified enormously by replacing a r e a l i s t i c intermolecular potential by a hard core repulsion. The molecules are then approximated by rigid spheres, and in the present model are assumed in addition to have no internal degrees of freedom. As in theMD model of a dilute gas discussed earlier (34_, 36), the molecular dynamics reduces then to free-flight motion between collisions* and results in a computational efficiency which makes i t possible to include chemistry at a l l . (2) Reactive transitions. Chemistry is introduced into the model by assigning reaction probabilities to members of a colliding pair at the instant of c o l l i s i o n . This process is handled conveniently, and in a theoretically consistent fashion, by assuming an activatio nel, and selecting in a followed. If a single reaction is possible, for example, the condition for reaction is simply that the relative kinetic energy of the colliding pair exceed the activation energy for the process. Once a particular reaction channel is selected, molecular identities and other properties of the reacting molecules are altered according to the laws governing the particular chemical transformation. All particles then move in free flight until the next c o l l i s i o n occurs and the channel-selection process is repeated. It is easy to see that these two components of the RMD simulation provide the information contained in the reaction probab i l i t y function P ( x , y ) . The molecular dynamics determines the type of c o l l i s i o n C|< and the elapsed time T , while the probability of reactive process R occurring upon c o l l i s i o n C^ is given independently in the second step. y

A Discrete Model of Cooperative Isomerization. The algorithm which implements the method of reactive molecular dynamics (RMD) is understood best in the context of a specific application. Therefore let us specialize now to the simple isomerization reaction (Eq. 2) to address the questions of "chemical nucleation" raised in that context. Only the main ideas are stressed here. A more detailed account of these studies appears in Ref. Z?. Consider a binary mixture of constituents A and B, and assume that the reversible isomerization is a collision-induced reaction. There are then four elementary reactive processes which may occur upon binary c o l l i s i o n in the system: (a)

A + A

+

A +B

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(6a)

18.

TURNER

Microphysics to Macrochemistry via Discrete Simulations (b)

k

i

245

B+ B

(6b)

A + A

(6c)

B+ A

(6d)

A + B Z (c) (d)

B+ B

->

Since the system is closed, the total number N E N/\ + N 3 is constant, and i t is easy to verify that in this case the determinist i c rate equation for reduces to the form of Eq. 4, where the constant N is absorbed into the pre-exponential factor kg of the latter equation. The sequence of c o l l i s i o n s , and hence a partial determination of reaction channel, is generated by the hard-sphere molecular dynamics algorithm. Fo is clear, and i t remain action, given the appropriate c o l l i s i o n , is satisfied. For type A-B collisions there are two possible channels from which to choose. Thus transition probabilities W (C^) are assigned which give the probability for reaction R to occur given a c o l l i s i o n of type C|<. Suitably normalized, these transition probabilities for the four channels of Eq. 6 are y

p

W ^ A - A ) - e" -nN /N B

e H

1 + e

b

^

£

o\ . e

/A

„

1 + e

A

e

-nN /N R

W

B

-

B

)

"

E

The probability of no reaction, given c o l l i s i o n C|<, is then W (C.) = 1 !

z W (CJ y=l p

K

(8)

K

Here i t is assumed that at most one reactive event per c o l l i s i o n can occur. Which of the possible events R occurs for a given collision is then determined by picking a uniform random number a from the unit interval. In general, therefore, i f y-l y y

Z W (C ) < a < Z W (C.) v=0 v=0 v

k

v

,

y = 1, 2,

W =0 Q

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

(9)

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then process R is selected. Once a particular channel has been selected, particle identities are altered according to the appropriate reaction path of Eq. 6. For A-A and B-B c o l l i s i o n s , a second random number is used to determine which of the (equally likely) molecules will react. In order to incorporate the macroscopic cooperativity parameter n into the dynamical model, consider what n means at the molecular level. Essentially n represents an average interaction energy, of a B molecule with other molecules of the same type, which tends to stabilize the given molecule with respect to conversion into the isomeric form A. Since intermolecular forces are of relatively short range, n contains the effects of interactions with nearest neighbors, next-nearest neighbors, e t c . , the strength of which decreases with distance. Hence nNg/N should properly reflect the influence of the local environment on the stability of a B molecule. This localization of the cooperativity is accomplished conveniently by introducing a "mean-field radius," R f. The instantaneous reactio becomes y

W ^ - e x p [-n(N /N) ] ~ mf B

A

B

(10)

R
which now includes only the fraction of B molecules within a mean-field radius of the reference B-molecule. In this relation n is identified with the deterministic quantity. In a dynamical simulation, therefore, R f can be determined by f i t t i n g the observed steady state (mole) fraction Ng/N to the deterministic curves (Figure 5). Such an empirical value of R ^ will necessari l y depend on the parameters which define the rigid sphere molecular dynamics model. If the interpretation given here is valid, however, R f should not depend on the chemical parameters e and n or on the total number N of molecules. In the simulations reported here, several simplifying assumptions have been made which have been relaxed in more recent studies: (a) The system is maintained isothermal by requiring that collisions are dynamically e l a s t i c , with no thermic effects due to heats of reaction, (b) The system is two-dimensional, which f a c i l i t a t e s greatly visualization of the chemical dynamics and reduces the complexity of the rigid-molecule dynamics, (c) The molecules are physically identical, with mass m and hardsphere (disk) diameter a m

m

m

Implementing the RMD Algorithm. Because the collisions are instantaneous in a rigid particle system, the "time to next c o l l i sion," t , is easily evaluated in terms of the positions and momenta of the N particles at any given time t (7). The particles are moved in straight lines to their new positions at time t + t , the momenta of the colliding pair are exchanged, and t reevaluated. Hence the dynamical evolution of the model N-body c

c

c

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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system may be followed exactly in a computer experiment. Obviously the computer storage and time requirements increase with N, so it is important to use as small a system as can reasonably simulate the phenomena of interest. We consider 100 rigid molecules in a two-dimensional region of area a and impose periodic boundary conditions to simulate a typical "cell" in a much larger system. Despite this generalization, the total number N of particles within the cell is preserved under periodic boundary conditions. If instead rigid (reflecting) boundaries were imposed, then the behavior of such small systems would be dominated by the walls, especially when spatially localized effects are important. The chemical evolution of the system would depend strongly on the size of the enclosing "box," and any connection to macroscopic chemistry would require extrapolation of small-system results. Undesirable size effects can be minimized, but not eliminated, with periodic boundary conditions. The possibility of boundary-related spatial correlations is then introduced, however, especially for very small "cells"; the significanc present model has yet to be established. The choice of dynamical i n i t i a l conditions is not c r u c i a l , since both thermalization of velocities and randomization of positions occur typically within a few collisions per particle. Hence a convenient square l a t t i c e i n i t i a l configuration is chosen, with uniform speed |v| = (2kgT/m)V2 corresponding to the given temperature T, and random direction vector. A "snapshot" of this i n i t i a l condition for a = 196a^ is shown in the f i r s t frame of Figure 6. Once chemical identities are assigned to the N molecules in the system, an evolution which simulates microscopic chemical kinetics can be followed, and appropriate dynamical and chemical data accumulated. In order to compare results with stochastic or deterministic theory (cf, Ref. 22 in the present case), i t is necessary to generate suitable "ensemble" averages by repeating each run a number of times (1_9, 22). This essential but expensive procedure will be unavoidable whenever the molecular dynamics depends on the chemistry (e.g., when "internal" energy is converted into kinetic energy via an exothermic reaction). A feature of the present computer model, by virtue of assumptions (a) and (c) above, is that particle motion can be decoupled from chemical processes. This means that each set of dynamical data can be generated without regard to chemistry and stored for repeated chemical simulations using different distributions of i n i t i a l chemical identities, different transition probabilities, or even entirely different chemical mechanisms. The result is an enormous savings in computer time, since the molecular dynamics remains the most time-consuming part of the RMD algorithm despite the use of hard-particle dynamics. Obviously, enough dynamical runs should be used to eliminate undesirable correlations due to repeated use of the space-time sequence of "common random numbers" generated in a single run. Even so, "ensemble" averages based on hundreds >

American Chemical Society Library

1155 16th St., N.W. Washington, D.C. 20036 In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER

248

MODELING O F M A T T E R

1 -00

Journal of Physical Chemistry

Figure 5. Steady states of the cooperative isomerization model, showing subcritical (rj = 1), critical ( = 4), and supercritical ( = 5) curves of mole fraction x as a function of the forward activation energy *. The deterministic transitions for = 5 are indicated by arrows; the dashed line denotes an "equal areas" construction which determines the unique equilibrium transition v

v

n

v

0.00

.80 1 .60 Z - 4 0 3-20 4.00

(22).

frame I

OOOO

oooo

OOOO

oooo oooo oooo oooo oooo oooo TiriE = 0.0000 frame 3

T I HE

T I HE = 101 .8445 frame 4

T I HE = 386.5326 Journal of Physical Chemistry

Figure 6. "Snapshots'* of an RMD simulation showing nucleation of the high-B phase, with = 5 and < = 2.348, beginning in the pure A state (x = 0 in Figures 5 and 7). Darkened disks denote B molecules; the short line inside each disk indicates the velocity vector. The mean-field radius B. = 2.50a is drawn in each frame. In frames 2-A it is centered at a colliding B molecule to indicate the fluctuation in mean-field cooperativity, especially evident in comparing frames 2 and 3 (22). v

B

mf

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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249

of chemical evolutions can be generated easily using common dynamical data from two or three runs and with no significant correlation effects. RMD Simulation of Chemical Nucleation ( 2 2 ) . A series of microscopic computer experiments was performed using the cooperative isomerization model (Eq. 2 ) . This system was selected for the t r i a l simulations for several reasons: F i r s t , only two chemical species are involved, so that a minimal number of particles is needed. Second, the absence of buffered chemicals ( e . g . , A and B in the Trimolecular reaction of the next section) eliminates the need for creation or destruction of particles in order to maintain constant populations (1_9, 2 2 ) . Third, the dynamical model of the cooperative mean-field interaction can be examined as a convenient means of introducing cubic or higher nonlinearity into molecular models based on binary collisions. Finally, the need for a microscopic simulation is most apparent for transitions between multiple macroscopic states. localized fluctuations is of obvious importance to the understanding of nucleation phenomena. As for the equilibrium vapor-liquid and liquid-solid transitions, detailed simulations at the molecular level should provide deep physical insight into chemical nucleation processes which is unattainable from theory, higherlevel simulation, or experiment. The RMD simulations used dynamical data generated by three molecular dynamics runs of 10^ collisions each. The runs involved 100 identical particles, in a box of side 16a, with periodic boundary conditions. The dynamical i n i t i a l conditions were identical except for the assignment of particle velocity directions using a different set of random numbers in each run. If = (N/2) /3 a i s the area occupied by N particles at close packing, then the close packing ratio r = / = 2.956 in these runs. Under these conditions the mean free path between collisions is £ - 1.14 a or roughly 1/14 of the box lenth. It was anticipated that the most sensitive feature of the macroscopic model to the cooperativity would be the location of the c r i t i c a l point ( n c > £ c > c ) ( 4 , 2 , . 5 ) . Hence the f i t of simulation results to the c r i t i c a l curve of Figure 5 was used to select the best mean-field radius for the system. The agreement obtained with R f = 2.50 a is shown in Figure 7. In this figure the curves are the deterministic results, and each data point represents the average of 11 RMD runs of 10^ collisions each, beginning with the same i n i t i a l mole fraction x = Ng/N, with d i f ferent random i n i t i a l assignment of particle identities. In the simulations of Figure 7 i t is apparent that most of the points studied on the metastable branches for n = 5 are in fact stable on the time scale of these computations. This should not be surprising, since the "size" of a destabilizing f l u c tuation is large near the "equal-areas" construction (dashed l i n e , Figures 5 and 7 ) , and decreases to microscopic dimensions only at 2

a

a

c p

r

x

=

m

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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MODELING O F M A T T E R

the marginal stability points of the deterministic curve (22, 51). Such large fluctuations should be extremely rare, and in particular require induction times much longer than the 10"^ sec duration of the 10 c o l l i s i o n sequence followed in these simulations. How then can one expect to see nucleation occur in a f i n i t e simulation involving only 100 particles? The only possibility is to prepare the system in a metastable state near the stability limit. In this situation the c r i t i c a l nucleus may be small enough that a destabilizing fluctuation would have a reasonable probability to occur ( c f . , the MC simulations of Ref. 52). Careful studies of localized fluctuation effects using microscopic chemical simulations are only beginning at the present tima Nevertheless, some i n i t i a l results have been obtained which are highly suggestive, although not yet definitive. With the same dynamical data and mean-field radius as before, several sets of RMD simulations were carried out in the neighborhood of the metastable states corresponding to n = 5. The sequence of frame evolution near the s t a b i l i t y limit of the lower metastable branch. The behavior of an entire series of 11 runs at e = 2.348 (e/n = 0.4696) is displayed in Figure 8. Here the Bmolecule population is plotted against c o l l i s i o n number for the f i r s t 10^ collisions. Within this relatively short time span a variety of types of evolution is found. All runs begin with 100 molecules in the form A. In this special case of a pure component f l u i d , both dynamical and chemical i n i t i a l conditions are the same for the entire set. In addition the sequence of dynamical events (collisions) is the same, so that the different results are due to the sequence of random numbers which determine the chemical evolution. In more general situations, the runs begin with a chemical mixture of fixed composition. The i n i t i a l identities of the molecules, assigned at random, are then different in each run, and this difference is reflected in the subsequent chemical evolution. Of the 11 systems studied (in Figure 8), only 2 spent no time at a l l in the metastable state at Ng - 16. These systems developed c r i t i c a l nuclei in the early generation of B molecules, as indicated in "snapshots" of their evolution, and proceeded directly to the stable steady state at NB ~ 89. The remaining systems a l l stabilized at the metastable state, 5 stayed there throughout the time shown in Figure 8, and one did not destabilize until after 5 x 10^ collisions had occurred. The other 4 developed c r i t i c a l nuclei during the time of observation as indicated in Figure 8, and evolved to the stable state. The different transition rates reflect two factors, the different sizes of the i n i t i a l supercritical nuclei, and the different effects of f l u c tuations, due to particle motion and reaction, on the growth of the nuclei. Due to the small numbers of molecules contained in a typical nucleus, these fluctuations tend to be large. In Figure 8 the fluctuations in NB have been suppressed so the 5

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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"average" evolution could be followed more easily. The range of excursions from the two steady states is indicated at the right border of the figure. In Figure 6 the sequence of events in a typical nucleation run is shown in the form of snapshots of the two-dimensional mixture. Beginning as before with a pure-A "phase" (frame 1), the system evolves f i r s t to the metastable state (frame 2), where i t remains for over 80,000 c o l l i s i o n s . By c o l l i s i o n 86,274 (frame 3), a supercritical nucleus has appeared. Thereafter this nucleus grows rapidly as the system evolves to the stable steady state (frame 4). This example was selected for display because i t exhibits a particularly "long-lived" metastable state and, together with Figure 8, illustrates the wide range of induction times observed in the formation of (super-) c r i t i c a l nuclei. A large number (hundreds) of RMD simulations have been performed near the limits of stability of the two metastable branches of Figure 7. Effects similar to those of Figures 6 and 8 were observed over a range o pected, a general feature of these studies is an increase in c r i t i c a l cluster size, and hence also in induction time, as values of e somewhat nearer the equal-areas line are chosen. Further details of the computer experiments, including quantitative features such as the distribution of induction times and the d i s tribution in size and composition of "chemical clusters," together with an applicable nucleation theory of chemical i n s t a b i l i t i e s , are subjects of active investigation at present, and will be reported elsewhere. Higher-Level

Simulations:

Stochastic Chemical Kinetics

Microscopic simulations, such as the RMD method discussed here, are not suitable for applications involving large numbers of particles, long simulation times, or chemical systems with widely differing reaction rates for different processes. The latter restriction in particular implies that many chemical systems of laboratory interest, which are governed at the deterministic level by "stiff" differential equations (48, 49), cannot yet be studied by the RMD method, even for short times in small systems. Such situations can be handled effectively in a higher-level simulation, however, in which individual microscopic processes are s t i l l followed, but no longer at the molecular level. In such a method (20, 21) only the chemical evolution (in a homogeneous volume) is simulated directly. Exactly as in the Markovian stochastic description of chemical kinetics, i t is assumed that details of the underlying molecular dynamics may be eliminated to good approximation, at least in locally homogeneous subsystems, by appropriate averages over the "microscopic stochastic variables" ( c f . , Ref.U). Hence a reduced description is adopted in which the species populations are followed, either globally in a spatially homogeneous system, or in locally homogeneous subvolumes or cells i f a global

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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description is not j u s t i f i e d . For a given system the relative rates of reaction and transport determine an upper limit on volumes which may be treated as macroscopically homogeneous. A lower limit is established by assumptions made in passing from the molecular to Markovian stochastic level of description (V7, 2 0 , 2 1 ) . Several methods for simulating the stochastic evolution of chemical systems have been employed in recent years (Ref. 21_ and references therein). Of particular interest is a stochastic simulation algorithm developed from the Markovian stochastic formulation of chemical kinetics (20^, 21_). Within this framework the transition probabilities for various kinetic processes take the general form W ( t ) 6 t = c h 6t

(11)

Here h is a combinatorial factor expressing the number of d i s tinct ways that the species necessary for an R reaction can be selected from the population t. The quantity c 6 t is then the "average probability that a particular combination of R reactant molecules will react in the next infinitesimal time increment 6 t " (21_, 2 2 ) . Usually c and h are simply related to the macroscopic rate constant k and the product of reactant species populations, respectively, as one would intuitively expect from macroscopic considerations. Turning now to the reaction probability function P(x,y) defined e a r l i e r , we can write an exact expression based on the Markovian stochastic theory applied to chemical kinetics (2(), 21_) y

y

y

y

y

y

P(x,y) = h c y

y

M exp ( - z h c x ) v=l v

(12)

v

Depending as i t does on the instantaneous values of c and h for all M reactions, P(x,y) is evidently a complicated function of the evolving chemical state of the system. Using this function, however, i t is possible to construct a rigorous algorithm for simulating the time-evolution of chemical systems to which the fundamental stochastic hypothesis applies (21_, 2 2 ) . The algorithm i n volves a proper random (Monte Carlo) selection, at the time t of an evolution, of quantities x and y which completely specify the next chemical event which will occur. With normalized transition probabilities obtained from Eq. 1 1 , the channel-selection proceeds as in the RMD simulations (cf, Eq. 9 ) . The appropriate time i n terval x is given in terms of a second uniform random number in the unit interval, 3, by the relation y

x = -W"

1

In 3 ,

WE Z y=l

h c u

u

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

y

(13)

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This selection process is then iterated, beginning from an i n i t i a l state of the system, as defined by species populations, to simulate a chemical evolution. A s t a t i s t i c a l ensemble is generated by repeated simulation of the chemical evolution using different sequences of random numbers in the Monte Carlo selection process. Within limits imposed by computer time restrictions, ensemble population averages and relevant s t a t i s t i c a l information can be evaluated to any desired degree of accuracy. In particular, reliable values for the f i r s t several moments of the distribution can be obtained both inexpensively and e f f i c i e n t l y via a computer algorithm which is incredibly easy to implement (21, 22), espec i a l l y in comparison to now-standard techniques for solving the s t i f f ordinary differential equations (48, 49) which may arise in the deterministic description of chemical kinetics (53.). Now consider briefly the essential features of a simple chemical model which illustrates well the attributes of stochastic chemical simulations. Symmetry-Breaking Instabilities: The Trimolecular Reaction. The "Brusselator" or trimolecular reaction is the simplest model which exhibits i n s t a b i l i t i e s that may be symmetry-breaking in space and/or time. Although i t does not represent an actual chemical reaction, i t is nevertheless the best-studied and most widely known theoretical model for chemical instability phenomena. Historically i t is the model on which the study of dissipative structures was begun by members of the Brussels. School of Thermodynamics (hence its popular name) a decade ago (44, 45, 46, 47). A detailed theoretical analysis of the Brusselator has been presented recently in a monograph by Nicolis and Prigogine (17). Relevant aspects of this development are reviewed in the following paragraphs: Without loss of generality the four reaction steps of the model are taken to be irreversible, thereby establishing the system i n f i n i t e l y far from equilibrium: ki A

•> k

B + X

2

~* k

2X + Y

+

X

+

Y +D (14) 3X

3

With A and B assumed to be in large excess (or buffered) so that they act as time-independent reservoirs, the deterministic rate equations for this system become (for convenience the rate con-

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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stants are set to unity) 2 | £ = A - BX + X Y - X + D 2

x

(15) = BX - X Y + D 2

y

8r

where r is the spatial coordinate for a one-dimensional system and Dx, Dy are Fick-type diffusion coefficients for intermediates X and Y , respectively. It is easy to see that this system admits a unique spatially homogeneous steady state solution (XQ = A, YQ = B/A), which is the continuous extension of the thermodynamic branch into the farfrom-equilibrium region. Using the familiar linear stability analysis, we find two types of situations in which this steady state may become unstable dent variable ( i . e . , A, Dx, Dy fixed), then the condition for an oscillatory instability is satisfied when B reaches B' =1 + A and

+

(D + D ) x

(16a)

Y

for a nonoscillatory instability when B reaches B" = (1 +

+

nrir

5

Y

!

¥ x ) JT

<

D

A

1 6 b )

where i is the length of the system and the range of permissible values for the wave number m depends on the boundary conditions. In order to understand better the nature of the i n s t a b i l i t i e s possible at these two points, consider the "critical" wave number m for disturbances which will produce instability at the smallest value of B. Minimizing the expressions in Eqs. (16a), (16b) with respect to m yields c

m' = 0

m

c =f D^) (

B^ = 1 + A B - - ( l

+

(17a)

2

A (

j*)

)

(17b)

In an actual chemical mixture there may be present spontaneous fluctuations having a wide range of wavelengths. As B increases, then, the instability which appears will correspond to the smaller of the two c r i t i c a l values Br. For an i n s t a b i l i t y at B<[, Eqs. (17a) imply that very long wavelength perturbations will grow and o s c i l l a t e , the system remaining spatially homogeneous. Before the instability point is

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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reached, fluctuations either decay monotonically or perform damped oscillations about the steady state. Beyond i n s t a b i l i t y , fluctuations are amplified, the system leaves the steady state, and evolves to an undamped periodic regime.The resulting dissipative structure is a unique (in this example) limit cycle which is asymptotically stable. That i s , an i n i t i a l point anywhere in the space X,Y tends to the same period trajectory (Figure 9). The important point is that the chemical reaction leads to coherent time behavior; i t becomes a chemical clock. It is clear from Eqs. 16 and 17 that the possible appearance of f i n i t e spatial inhomogeneities involves a "cooperation between chemistry and transport. Indeed, whenever the rates of diffusion and chemical reactions are "comparable," short wavelength disturbances may grow. For an i n s t a b i l i t y at B£, for example, the system amplifies a fluctuation of "critical" wave number mc and u l t i mately reaches a stable nonuniform steady state (see Figure 10). The behavior of thi been the subject of extensiv f i r s t spatial dissipative structures were predicted using the Brusselator equations (17, 44, 45, 46, 47). In general, of course, the length of the system will not be an integral multiple of the c r i t i c a l wavelength obtained from Eqs. (17b). This leads to an indeterminacy in the nature of the spatial dissipative structure which bifurcates from the homogeneous steady state at B£. The system will select that wavelength, compatible with the boundary conditions, which l i e s nearest the wavelength of the particular disturbance which destabilizes the i n i t i a l steady state. Once one or another dissipative structure is established, however, i t is then stable with respect to small fluctuations, and hence in i t s wavelength possesses a primitive "memory" of the fluctuation from which i t originated. Other than the two simplest p o s s i b i l i t i e s treated here, the Brusselator also admits more complicated situations, including time-independent, oscillatory, and travelling wave solutions of the governing macroscopic equations. When nonuniform distribution of the reservoir variable A is permitted, moreover, spatially localized structures of several types have been realized. The implications of a l l these predictions for self-organizing systems in chemistry, biology, . . . are discussed in Ref. 1_7, as are the numerical (deterministic) simulations which verify the theoretical predictions. 11

Spontaneous Fluctuations at Chemical Instabilities via Stochastic Simulations"! Let us now consider an application of stochastic simulation methods to the Trimolecular reaction. Here we consider a few realizations of the underlying stochastic process in order to i l l u s t r a t e the main ideas. The i n i t i a l investigation focuses on the size-dependence of fluctuations near the homogeneous transition to limit cycle oscillations. The deterministic system is defined according to Eqs. 14-17, with A = 1 and k = 1 y

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I .00

7= 5 .80

• 60 Journal of Physical Chemistry

BO

Figure 7. Average composition (x, ) obtained from RMD simulations with R^/ = 2.50CT. Solid curves give the deterministic steady states, filled circles denote computer-experimental steady states which are stable during the time of observation, while open cricles denote metastable steady states of the computer model (22). {

.40

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• 20

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• •

.47

•

•

•

;

.49

.51

; .55

.53


100

98

L R

Journal of Physical Chemistry

/

/

75

/ V

60

Figure 8. Evolution of N in a set of 11 runs using common dynamical data. 4 0 Same parameters and initial conditions as Figure 6. The first 10% of the full 20 ^-collision simulation is shown. Fluctuations about the trajectories have been suppressed, but the range of fluctuations about the two steady states is indicated at right (22).

E

/

•

/

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r

/

/

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r /

/

Y

_A.?» L ._ -_: C

J

K

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5000 Collisions

Figure 9. Approach to stable limit cycle from unstable steady state S and from other initial conditions (A = I, B = 3,

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o r y = [1,4]. In a spatially homogeneous system, then, the transition to bulk oscillations will occur at the c r i t i c a l point given by Eq. 17a, or = 2. The object of the simulations is to study fluctuations in a stochastic version of this system, for values of B in the s u b c r i t i c a l , c r i t i c a l , and supercritical regions, and to characterize the fluctuations as a function of the size of the system. If A denotes a number density (or molar concentration), then N/\ E A • V (or A • a), the number of molecules of type A is a direct measure of the system size. (The system size factor a = NQV for molar concentrations, where NQ is Avogadro's number.) Hence the simulation strategy will be to pick a size Nfl, which also determines the steady-state X population NX = N/\, and vary B ( i . e . , NB) through the region of interest. The results of stochastic simulation for N/\ = 500 are d i s played in Figure 11. Here are plotted the populations Nx, Ny against time in a, c, e, and the Nx - Ny phase portrait in b, d, f, for three values of B indicated on each plot. figures represent time-dependent averages, taken over times long compared to the time between reactive events, but short on the time scale of any macroscopic evolution. Hence a high frequency (molecular) component has been averaged out of the raw population data. In the far subcritical region (B = 1; Figures 11a,b) the system remains near the point (N$,NQ), but its random evolution tends to smear out the deterministic steady state. In the time trace fluctuations are seen to be small. As B increases toward B£, the random motion about the steady state increases in amplitude u n t i l , at the c r i t i c a l point (B = 2; Figures 11c,d), occasional large amplitude quasi-oscillations appear in the otherwise chaotic motion about (N^,NQ). Beyond B£, the oscillations gradually i n crease in coherence, but remain relatively erratic until the amplitude of oscillation greatly exceeds the background level of thermal fluctuations. In the far supercritical region (B = 3; Figures l l e , f ) , a regular pattern is evident, with large amplitude relative to fluctuations. Nevertheless, even here a degree of incoherence remains, a direct manifestation of the f i n i t e numbers effect common to a l l discrete simulations. Following the series with N/\ = 500, further simulations were performed with N/\ values of 100, 1000, and 5000. The results of all the homogeneous simulations are collected in Figure 12. Here is plotted the average peak-to-peak amplitude of excursions in the X-population, normalized by the size parameter N/\ to give an i n tensive quantity for purposes of comparison. This "reduced Xamplitude" is plotted as a function of B over a wide range which includes the c r i t i c a l point Be = 2. Each point represents an average over ten runs which d i f f e r only in the sequence of random numbers used in the channel-selection process. Error bars for several N/\ = 100 points indicate the dispersion in the individual runs. As one would expect, this dispersion decreases with f

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Figure 10. Stationary distribution for 1(K) boxes with fixed boundary concentrations (X = 2, Y = 2.62, D = .0016, D = .008, A = 2, B = 5.24, k = J, T

y

M

(6)

B = I.O

1000 2000 X (nunb.r ) 3000 2500

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3000

B = B =2.0 C

Z000 °1500 x 3 -1000

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1000 X

2000

(nuMb.r

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)

Journal of Physical Chemistry

Figure 11. Time traces of the X, Y populations (a,c,e) and their associated phase portraits (b, d, f). The data are from stochastic simulation of the homogeneous trimolecular reaction, in the subcritical (a, b), critical (c, d), and supercritical (e,f) regions. A = 2, k = 1 = [1,4]), B' -= 2; N = 500 (22). M

c

A

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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0.0

1.0

2.0

3.0

[ B] Journal of Physical Chemistry

Figure 12. Reduced N amplitude near the second-order transition to homogeneous oscillations in the trimolecular reaction, as a function of the system size, N . The order parameter value at each experimental point is an average over 10 simulations, the error bars for N = 100 indicating the dispersion about the mean. Same macroscopic parameters as Figure 11 (22). x

A

4

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increasing N/\. Qualitatively, the second-order character of the transition to homogeneous oscillations is already evident in Figure 12. Below B , the f i n i t e values of the order parameter, as measured by ANx/N ,reflect the effect of fluctuations about the homogeneous steady state. For B < 1, the fluctuations are found to obey the •'square-root rule," to several decimal places accuracy, as would be expected far from a c r i t i c a l point. Similarly, for B > 2.5, fluctuations about the macroscopic evolution behave in the "classical" manner. Near B£, however, this simple size-dependence of fluctuations breaks down. Finally, the order-parameter data are seen to approach a definite limiting curve as the system size increases, indicating increased coherence and long-range temporal order. According to the phase transition analogy, we would expect to find perfect temporal coherence ( i . e . , a chemical clock) in the limit of a macroscopic homogeneous system. Further stochastic simulation studies now in progress are concerned with fluctuatio systems (e.g., limit cycle o s c i l l a t i o n s , combustion and explosions) and at the transition to spatial dissipative structure ( c f . , Figure 10). In the latter case, for example, stochastic simulations verify the existence of c r i t i c a l long-range spatial correlations predicted in a stochastic theoretical study of the model ( c f . , Ref. 17_). c

A

Concluding Remarks As we have seen, the need for chemical simulations at a v a r i ety of levels is quite clearly demonstrated in the broad class of systems in which molecular and supermolecular interactions lead to macroscopic cooperativity, self-organization, and evolution. Although this class extends far beyond the domain of "classical" chemistry as well, i t nevertheless covers the chemical spectrum, from combustion and explosions in gaseous mixtures to biochemical regulation of metabolism and biosynthesis, and from adsorption/ chemisorption and heterogeneous catalysis to chemical o s c i l l a tions, spatial structures, and travelling chemical waves in condensed phase chemistry and biology. It is evident from the proliferating literature in this f i e l d (cf, Ref. V7) that application of macroscopic and s t a t i s t i cal theoretical methods to chemical i n s t a b i l i t i e s is far more d i f f i c u l t in general than for their equilibrium counterparts. In particular, the use of any analytical theory to study typical chemical systems of laboratory importance is out of the question, so i t is precisely in this connection that discrete simulation methods may ultimately make their greatest contribution. In addition, a number of important fundamental questions regarding nonequilibrium phase transitions are beyond resolution with available stochastic methods, and indeed require a molecular picture. Here again the answer is not readily available from

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theory. As one might expect from studies of analogous problems in equilibrium phase transitions and c r i t i c a l phenomena, nonequilibrium phase transitions present great d i f f i c u l t i e s in both microscopic theory and laboratory experiment. In both experimental and theoretical studies, therefore, simulations involving actual molecular dynamics, or the higher level reactive molecular dynamics, may provide the necessary i n formation at a not unreasonable expense. When absolute molecular detail is not required, or can be incorporated into average quantities to parameterize a higher-level simulation, then stochastic simulations provide a powerful yet flexible technique which should be applicable to a wide variety of problems. Because they are limited neither by number of variables nor by mechanistic complexity, such methods afford a practical means of treating highly complicated systems. Moreover, the problems of "stiffness" which plague deterministic methods do not arise in stochastic simulations. Consequently, for some problems, stochastic simulations may actually out-perfor scopic differential equations, particularly when large numbers of chemical species governed by s t i f f kinetic equations must be treated in two or three space dimensions. At present this conjecture must be qualified by the practical limitation of discrete event methods to relatively small systems for relatively short simulation times (e.g., up to 10^ molecules for 10^ to 10° events). This paper has focused on two recent computer methods for discrete simulation of chemical kinetics. Beginning with the realization that truly microscopic computer experiments are not at all feasible, I have tried to motivate the development of a hierarchy of simulations in studies of a class of chemical problems which best i l l u s t r a t e the absolute necessity for simulation at levels above molecular dynamics. It is anticipated (optimistically!) that the parallel development of discrete event simulations at different levels of description may ultimately provide a practical interface between microscopic physics and macroscopic chemistry in complex physicochemical systems. With the addition to microscopic molecular dynamics of successively higher-level simulations intermediate between molecular dynamics at one extreme and differential equations at the other, i t should be possible to examine e x p l i c i t l y the validity of assumptions invoked at each stage in passing from the molecular level to the stochastic description and f i n a l l y to the macroscopic formulation of chemical reaction kinetics. With key algorithmic elements drawn from familiar MD and MC methods and from a Markovian stochastic theory of microscopic rate processes, the two methods discussed here greatly idealize important microscopic features of the systems under study, and hence represent only the f i r s t few steps in the overall program. In this sense they may be reasonably viewed more as an extension "downward" from the macroscopic description than as a faithful representation of microscopic chemical dynamics. Appropriately,

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therefore, the i n i t i a l applications have been to questions raised in deterministic (and stochastic) theories of chemical kinetics. As we have seen already, however, atomic and molecular detail can be incorporated into computer simulations by performing suitable averages over selected microscopic degrees of freedom. In this way effective force laws, for one example, or microscopically derived reactive transition probabilities ( i . e . , "rate constants"), for another, should permit simulations at a variety of levels of microscopic d e t a i l . Abstract For the broad class of systems in which molecular interactions lead to macroscopic collective behavior, i t is often the case that neither microscopic theory nor laboratory experiment can provide unambiguous interpretation of observed phenomena. Faced with such an impasse, one may reduce the gap between theory and experiment by means of discret puters. Such methods have long been invaluable in the study of classical f l u i d s , for example, and in various forms are used in all disciplines which must treat complex interactions among large numbers of individual elements. Within chemistry the physical properties of solvents, surfaces, and complex molecules are already studied by means of computer "experiments," and further applications cover the chemical spectrum from combustion and explosions to polymerization and biochemical regulation, and from sorption and heterogeneous catalysis to chemical i n s t a b i l i t i e s , self-organization, and evolution far from equilibrium. At the level of molecular dynamics, reactive collisions are rare dynamical events, so complete microscopic simulations of even simple chemical processes are not feasible. To the extent that the i n fluence of nonreactive processes on chemistry can be either ignored or treated s t a t i s t i c a l l y , however, i t is possible to develop higher-level simulations in which only events of direct chemical interest are followed. In this way a hierarchy of d i s crete simulations can span the gap between molecular dynamics and chemical kinetics, with additional computational efficiency gained at the expense of unwanted molecular d e t a i l , providing ultimately a practical interface between microscopic physics and macroscopic chemistry. Acknow!edgement I wish to thank Professor I. Prigogine, Drs. J. Hiernaux and D. T. Gillespie, and Messrs. S. Kelkar and M. Malek-Mansour for stimulating discussions on the many facets of interfacing discrete simulations and chemical kinetics. I thank especially Dr. Hiernaux and Mr. Kelkar for their efforts in computational aspects of the RMD and stochastic simulations. This work was supported in part by the Robert A. Welch Foundation.

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Literature Cited 1.

2. 3. 4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14. 15. 16. 17. 18. 19. 20. 21. 22. 23. 24. 25. 26. 27. 28. 29. 30.

B.J. Alder and T.E. Wainwright, "Proceedings of the International Symposium on Transport Processes in Statistical Mechanics" (Brussels, Aug. 27-31, 1956; I. Prigogine, Ed.) Interscience, New York, 1958, pp. 97-131. B.J. Alder and T.E. Wainwright, J . Chem. Phys., 31, 459 (1959). A. Rahman, Phys. Rev., 136, A405 (1964). L. Verlet, Phys. Rev., 159, 98 (1967). A. Rahman and F.H. S t i l l i n g e r , J . Chem. Phys., 55, 3336 (1971). D.L. Bunker, Methods Comp. Phys., 10, 287 (1971). W.W. Wood, in "Fundamental Problems in Statistical Mechanics," Vol. 3 (E.G.D. Cohen, Ed.) North-Holland, Amsterdam, 1975, pp. 331-388. J . R . Beeler, J r . , Adv. in Materials Research, 4, 295 (1970). W.G. Hoover and W.T. Ashurst, in "Theoretical Chemistry-Advances and Perspectives, son, Eds.) Academi W.W. Wood and J . J . Erpenbeck, Ann. Rev. Phys. Chem. 27, 319 (1976). F.T. Wall, L.A. H i l l e r , J r . , and J . Mazur, J . Chem. Phys., 29, 225 (1958). F.T. Wall, L.A. H i l l e r , J r . , and J. Mazur, J . Chem. Phys., 35, 1284 (1961). D.L. Bunker, J . Chem. Phys., 37, 393 (1962). J . C . Keck, Disc. Faraday Soc., 33, 173 (1962). N.C. Blais and D.L. Bunker, J . Chem. Phys., 37, 2713 (1962). N.C. Blais and D.L. Bunker, J . Chem. Phys., 39, 315 (1963). G. Nicolis and I. Prigogine, "Self-Organization in NonEquilibrium Systems," Wiley-Interscience, New York, 1977. J . Portnow, Phys. Letters, 51A, 370 (1975). P. Ortoleva and S. Yip, J . Chem. Phys., 65, 2045 (1976). D.T. G i l l e s p i e , J . Comp. Phys., 22, 403 (1976). D.T. G i l l e s p i e , J . Phys. Chem., 81, 2340 (1977). J . S . Turner, J . Phys. Chem., 81, 2379 (1977). N. Metropolis, A.W. Rosenbluth, M.N. Rosenbluth, A.H. Teller, and E. T e l l e r , J . Chem. Phys., 21, 1087 (1953). W.W. Wood, in "Physics of Simple Liquids" (H.N.V. Temperley, J . S . Rowlinson, and G.S. Rushbrooke, Eds.) North-Holland, Amsterdam, 1968, pp. 115-230. G.A. Bird, Ann. Rev. Fluid Mech., 10, 11 (1978). S. Glasstone,K . J . Laidler, and H. Eyring, "Theory of Rate Processes," McGraw-Hill, New York, 1941. C H . Bennett, in "Algorithms for Chemical Computation" (R.E. Christoffersen, Ed.) ACS Symposium Series, Vol. 46, American Chemical Society, Washington, D.C., 1977, pp. 63-97. S.A. Adelman and J . D . Doll, J . Chem. Phys., 61, 4242 (1974). S.A. Adelman and J . D . Doll, J . Chem. Phys., 62, 2518 (1975). J . D . Doll, L.E. Myers, and S.A. Adelman, J . Chem. Phys., 63, 4908 (1975).

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

COMPUTER MODELING OF MATTER

264

31.

S.A. Adelman and B.J. Garrison, J. Chem. Phys., 65, 3751 (1976) . 32. J.D. Doll and D.R. Dion, J. Chem. Phys., 65, 3762 (1976). 33. F.W. deWette, R.E. Allen, D.S. Hughes, and A. Rahman, Phys. Letters, 29A, 548 (1969). 34. W.C. Harrison and W.C. Schieve, J. Stat. Phys., 3 35 (1971). 35. H.W. Harrison, W.C. Schieve, and J.S. Turner, J. Chem. Phys., 56, 710 (1972). 36. H.W. Harrison, "A Molecular Dynamics Study of Dimer Formation," Ph.D. Dissertation, University of Texas, Austin, 1972. 37. H.W. Harrison and W.C. Schieve, J. Chem. Phys., 58, 3634 (1973). 38. W.C. Schieve and H.W. Harrison, J. Chem. Phys., 61, 700 0(1974). 39. M. Synek, W.C. Schieve, and H.W. Harrison, J . Chem. Phys., 67, 2916 (1977). 40. W.H. Zurek and W.C. Schieve, Comp. Phys. Commun., 13, 75 (1977). 41. W.H. Zurek and W.C (1978). 42. W.H. Zurek and W.C. Schieve, Phys. Letters, A (in press, 1978). 43. J.O. Hirschfelder, C.F. Curtiss, and R.B. Bird, "Molecular Theory of Gases and Liquids," John Wiley, New York, 1954. 44. I. Prigogine and G. Nicolis, J. Chem. Phys., 46, 3542 (1967). 45. R. Lefever, G. Nicolis, and I. Prigogine, J. Chem. Phys., 47, 1045 (1967). 46. I. Prigogine and R. Lefever, J. Chem. Phys., 48, 1695 (1968). 47. R. Lefever, J. Chem. Phys., 49, 4977 (1968). 48. C.W. Gear, Commun. Am. Comput, Mach., 14, 185 (1971). 49. D.D. Warner, J. Phys. Chem., 81, 2329 (1977). 50. T.L. H i l l , "Introduction to Statistical Thermodynamics," Addison-Wesley, Reading, Mass., 1960. 51. I. Prigogine, R. Lefever, J.S. Turner, and J.W. Turner, Phys. Letters, 51A, 317 (1975). 52. A.I. Michaels, G.M. Pound, and F.F. Abraham, J. Appl. Phys., 45, 9 (1974). 53. Proceedings of a "Symposium on Reaction Mechanisms, Models, and Computers," 173rd Nat. Mtg. of the Amer. Chem. Soc., March 20-25, 1977, published in J . Phys. Chem., 81, no. 25 (entire issue)(1977). 54. M. Rao, B.J. Berne, and M.H. Kalos, J. Chem. Phys., 68 (4), 1325 (1978). RECEIVED September 7, 1978.

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

INDEX Brusselator 253 A "Brute force" method 87 Algorithm(s) Green's function Monte Carlo 220 "leap-frog" 2 C metropolis 159 optimization of sampling 159 Cation(s) alkali metal 210, 212 predictor-corrector 2 monatomic 206 rejection sampling 16 sampling 16 function 207,208 for sampling preferentially near 66 the solute 165 Center-center distance 255 singularity-free 62 Clock, chemical Alkali halides 97 Cluster "critical" 236 Alloys 101 distributions, physical 236 Anion(s) "supercritical" 236 halide 211,212 173 monatomic 208,209 Coefficients, virial Computer storage 153 -water radial distribution field 86 function 208,209 Conductivity, high Conformation defects in polyethylene Aqueous solutions, structure of liquid chains 140 water and dilute 191 Array processor I l l Correlations, critical long-range spatial 260 floating point systems 190-L 112 "Critical" cluster 236 word length on precision, effect of 123 phenomena 231, 261 Atom-atom potential 77, 80 wave number 254 Atoms dissolved in water, mean force Crystal(s) of two noble gas 32 molecular 101 Axilrod-Teller packing in polyethylene 140 fluid 182,186,187 properties in polymer 137 force 182 quantum 219 three body 184 260 potential 174,175,183 Cycle oscillations, limit triple dipole potential 176, 177,186 D B

Banded-matrix approximation Bead chains, eightBead-and-spring model Ben-Nairn Stillinger (BNS) potential Benchmarks general Monte Carlo results of Born-Mayer-Huggins Potential Born-Mayer repulsions Bottlenecks Bound states (dimers) Bragg-Williams (meanfield)model Brownian motion

141 134 125 35 118 118 116 2 97 233 235 242 126

Density harmonic coefficient 82 -orientation profile 83 profile . 80, 82 Differential equations, "stiff" 251 Dimers (bound states) 235 Dimer, formation and destruction 237 Discrete-event simulation 233 Discs, hard 192 Distance, center-center 66 Distance, site-site 66 Distribution function anion-water radial 208, 209 for binding energy, quasicomponent 196, 201

267 In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

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Distribution function (continued) cation-water radial 207, 208 methane-water radial 202 oxygen-oxygen radial 197 probability 168 quasicomponent . 197,199, 203, 210-212 radial 173,185, 186 solute-solvent radial 167,168 Distributions, physical cluster 236 Dynamic and equilibrium properties . 125 Dynamics, many-body 233 E

Energy calculations, conformational 137 configurational internal 69 free 173 function, Kistenmacher, Popkie, and Clementi (KPC) "simple" potential 4 internal 173 kinetic 23-25 minimization 138 partial molar internal 201 potential 19,148 and pressure 184 quasicomponent distribution function for binding 196, 201 surface, Hartfree-Fock 40 thermodynamic internal 196 total 4,19,20 Equilibrium data 129 Equilibrium properties 185 dynamic and 125 Ewald method 88 Ewald summation 22 F

Fletcher-Reeves method 137 Floating Point Systems 190-L Array Processor 112 Fluctuations 239 Fluid(s) Axilrod-Teller 182,186,187 Lennard-Jones 73,182,186,187 liquid-vapor surface of molecular .. 76 molecular dynamics simulations of simple 172 Monte Carlo calculations on 159 "pair theory" 172 two-dimensional Lennard-Jones .... 174 Fluoride 208 Force 148 Axilrod-Teller 182 dispersion 97 Lennard-Jones 182

Force (continued) potential, shifted 146 three body 173 Axilrod-Teller 184 of two noble gas atoms dissolved in water, mean 32 two- and three-body 182 Formalism, Hastings 161 Fourier transform method 88 Freezing of simple systems Ill Fused salt systems 88 G

Gas(es) electron molecular dynamics in dilute solids, rare Green s function Monte Carlo algorithm

219 234 97 220

H Halides, alkali Hard-sphere dynamics Hartfree-Fock energy surface Hastings formalism He-3 He-4 structure functions for Host overhead typical values for Hydronium ion, trihydrated

97 244 40 161 219 219 227 120 121 51

I Instabilities chemical 231,239 and nonequilibrium phase transitions 233 nonoscillatory 254 symmetry-breaking 253 and transitions, nonequilibrium 231 Integration schemes, predictorcorrector 155 Interactions, molecular dynamics with three-body 178 Interactions, three-body 172,174 Interference effect 103,105 Ionic interactions, molecular dynamics simulations of liquids with 1 Isomerization model, cooperative 248, 249 Isomerization reaction, cooperative 242

J Jonesfluid,two-dimensional, Lennard-

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269

INDEX

Models studied, root mean squared deviations for 67 Molecular crystals 101 dynamics 231 (CMD), conventional 178 in dilute gases 234 (MD) method 94 program 155 L reactive 241, 243 simulations 144 Lattice-model polymer chain 125 of liquids with ionic inter"Leap-frog" algorithm 2 actions 1 Lennard-Jones of methane 62 fluid 73,182,186,187 of simple fluids 172 two-dimensional 174 with three-body interactions 178 force 182 liquids 144,151 particles 192 site-site potentials for 145 spheres 32 mechanics method 137 system 98 147 Limit cycle 255 Molecules, spherical "Link" system 8 LiOH, hydrate of 51 algorithm, Greens function 220 Liquid(s) benchmarks 118 with ionic interactions, molecular calculations on fluids 159 dynamics simulations of 1 computer simulation 202 molecular 144, 151 loops, inner 115 nucleation, vapor236 methods 111,219 phase transition, vapor236 program, N-particle 119 quantum 219 simulation(s) 50, 56, 57, 72 -vapor interface, structure of a 72 of water 29 -vapor surface of molecular fluids 76 studies 191 Lithium 207 techniques 125 cation 126 dihydrated 47 Motion, Brownian 232,233 hydrated 35 Multiple time scales four-hydrated 47, 51, 52 Multiple time step (MTS) method 144,147,179 trihydrated 47 linear 183 with water, complexes of the 44 -water potential calibration 40 Loops, inner Monte Carlo 115 N Nernst-Einstein equation 87 M Newton-Raphson method 137,139,142 Marginal stability 243 Nonadditivity, previous work on 173 Matrix Nucleation o, optimization of step acceptance . 162 chemical 244,249 approximation, banded 141 homogeneous 234 Q, optimization of transition 162 RMD simulation of chemical 249 Matter, neutron 219 vapor-liquid 236 Matter, nuclear 219 Nucleus 243 Melting of simple systems Ill "supercritical" 243 Metals 101 Metastable 243 O Methane 203 83 dilute aqueous solution of 200 Orientation profile, density239 molecular dynamics simulation of 62 Oscillations, chemical 260 stereographic view of 204, 205 Oscillations, limit cycle -water radial distribution 202 Oxygen-oxygen radial distribution .... 196 function 197 Metropolis algorithm 159 K Kinetics, Markovian stochastic description of chemical 241 Kistenmacher Popkie Clementi potential function 40, 42 Kramers-Kronig dispersion relations . 92 Kubo type relations 87

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270 P

"Pair theory" of fluids 172 Parameter sets studied, potential 66 Particle trajectories 188 Pechhold kink 140, 142 Peskun's results 164 Phonons 94, 106, 107 Polarization, effect of 102 Polarization Model 35, 39 Polyethylene chains, conformation defects in 140 Polymer chain, lattice-model 125 chain, single linear 126 system, multichain 125 Potassium 208 Potential function(s) 42 Kistenmacher Popkie Clementi 42 Precision, effect of AP word length on 123 Predictor-corrector algorithm 2 Predictor-corrector integration schemes 155 Pressure 69 internal energy and 184 Program development 114 Proton, quadrihydrated 51 Q Quantum crystals Quantum liquids

219 219 R

Radial distribution, oxygen-oxygen 196 Reaction, trimolecular 253 Reactive molecular dynamics (RMD) 240 Rejection sampling algorithm 166 Relaxation method 137, 142 hybrid Newton Raphson 139 Relaxation times for vector end-to-end length 133 Reneker twist 140 RMD simulation^) 248 of chemical nucleation 249 Root mean squared deviations for models studied 67 Rouse model 126 S

Salts, molten Shifted force potential Simulations averages obtained during chemical hierarchy of of liquids with ionic interactions, molecular dynamics

1 146 84 260 261 1

MODELING O F M A T T E R

Simulations (continued) methods, discrete 260 microscopic 249 chemical 241 stochastic 241, 252, 260 Singularity-free algorithm 62 Site-site distance 66 Site-site potentials) 77 for molecular liquids 145 Smooth twist 140 Sodium 207 Solids, collective modes in 94 Solute, algorithm for sampling preferentially near the 165 Solute-solvent radial distribution function 167,168 Solutions, aqueous 14 Speed, computing 154 Spheres 192 Spring model, bead-and125 Steepest descent method 137 Step methods, multiple time 144, 147 "Stiffness," numerical 241 Stochastic description of chemical kinetics, Markovian 241 Stochastic simulation 241 Storage, computer 153 Storage requirements, computing time and 153 Structure dissipative 239,255 factor, dynamical 95 spatial dissipative 255, 260 "Supercritical" cluster 236 Symmetry-breaking instabilities 253 T Teller force, three-body Axilrod184 Teller potential, Axilrod 174-177,183,186 Thermodynamic branch 239 Time, computing 153 Time and storage requirements, computing 153 Transition (s) chemical instabilities 233 equilibrium phase 261 first-order chemical phase 242 nonequilibrium instabilities and 231 nonequilibrium phase 231, 233, 239, 261 phase 231 reactive 244 state theory 233 vapor-liquid phase 236 Trimolecular reaction 253 Truncated potentials 145

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

271

INDEX

v Van Hove function 95 Vapor interface, structure of a liquid72 -liquid nucleation 236 -liquid phase transition 236 surface of molecular fluids, liquid76 Vector end-to-end length, relaxation times for 133 Velocity autocorrelation functions 10, 13, 18 Verlet algorithm 88

Water (continued) liquid Monte Carlo simulation of potential calibration, lithium cationradial distribution function, anion208, function, cation207, methanestatistical state of liquid 199, Wave number, "critical" Waves, chemical Williams (mean field) model, Bragg-

195 29 40 209 208 202 202 254 239 242

W Water complexes of the lithium cation with and dilute aqueous solutions, structure of liquid

X

44 191

Xenon atoms

In Computer Modeling of Matter; Lykos, P.; ACS Symposium Series; American Chemical Society: Washington, DC, 1978.

32